WASHINGTON — Of all the “bitter disappointments” he had as president, Harry Truman once wrote, the “one that has troubled me most, in a personal way,” was the failure to enact a national program that would assure all Americans “a full measure of opportunity to achieve and enjoy good health.”

More than 75 years after Truman first proposed universal coverage, Democrats are still chasing his dream. If President Biden’s social policy bill becomes law, they will make major strides toward fulfilling it.

An estimated 3.4 million Americans would gain health insurance as a result of the legislation, which passed the House last month but faces a tough road in the 50-50 Senate. Senator Chuck Schumer of New York, the majority leader, said Tuesday that his goal is to have it pass before Christmas.

The bill would expand health care access for children, make insurance more affordable for working-age adults and improve Medicare benefits for the disabled and older Americans. Separately, its health provisions are a “piecemeal of incremental changes,” said Caroline Pearson, a senior vice president for health care at NORC at the University of Chicago, a nonpartisan research organization.

But taken together, these policies represent the biggest step toward universal coverage since the passage of the Affordable Care Act in 2010.

“This is a moment of extraordinary opportunity for improving health policy and improving the health coverage that people get,” said Stan Dorn, director of the National Center for Coverage Innovation at the advocacy group Families USA.

House Democrats, facing near-unanimous opposition from Republicans and pushback from more centrist members of their own party, failed to include some of the splashy health proposals that were discussed early in the negotiations over the package.

Medicare will add hearing coverage, but not a vision or dental benefit. The government will not gain the ability to bargain down the prices of hundreds of prescription drugs, though it will be able to lower the prices for 20 each year in the Medicare program.

If the measure passes, the United States will retain its patchwork system, where people obtain different health coverage depending on where they live, what they earn, where they work and how old they are.

Even so, Ms. Pearson said, the legislation “is one of the biggest steps toward patching the holes” in the system.

As a group, the health care provisions will cost $330 billion over the next decade and come with compensating health savings of $325 billion, according to an analysis of Congressional Budget Office data by the Committee for a Responsible Federal Budget. But that balance is slightly misleading: The parts that save money are designed as permanent, while several new coverage provisions would expire after 2025.

Even with the changes, the Congressional Budget Office estimates that more than 27 million people would remain uninsured, including many undocumented immigrants, whom the bill does not assist. Many of the remaining uninsured would be eligible for expanded Obamacare subsidies or Medicaid, but are not expected to sign up.

Here are some of the programs — and people — the legislation would affect.

Christina Ruiz had plenty to worry about when she gave birth two months early, in August 2020.

Her infant daughter had to spend five weeks in the neonatal intensive care unit while Ms. Ruiz, 34, dealt with her own postpartum complications. She developed high blood pressure, and the stitches on her C-section incision began to unravel three weeks after delivery.

One thing Ms. Ruiz did not have to worry about: medical bills. She had enrolled in Medicaid early in her pregnancy, and it fully covered both her and her daughter’s costs.

But while Medicaid has become a major source of health coverage during pregnancy — about 40 percent of the country’s babies are born to mothers who receive the coverage — it ends 60 days after delivery. Researchers say this is an especially problematic time for women to lose health insurance, when they are still at high risk of postpartum complications.

The United States has the highest maternal mortality rate in the developed world, and about 12 percent of such deaths happen more than six weeks after delivery. A lack of insurance may play a role.

The social policy bill would provide Medicaid to new mothers for a full year after delivery instead of just two months, allowing more time to address postpartum medical issues that can surface later.

The Century Foundation estimates the provision would extend coverage to about one million women over the next decade.

The legislation would also expand coverage for children, by permanently funding the Children’s Health Insurance Program, which covers 10 million low- and middle-income children, and by making it harder for children to lose Medicaid coverage because of paperwork errors or fluctuating family income.

The postpartum coverage could help Ms. Ruiz, who is now pregnant with her second child.“It makes all the difference,” she said, “not having to worry about health care bills.”

Tim Floyd of Guntown, Miss., was working construction jobs in 2012 when he noticed numbness in his foot. It was neuropathy, a sign of diabetes. But he was uninsured and could not afford a doctor visit.

“If you are having to pay $60 out of pocket, you go, ‘Well, it’s not exactly right, but it’s not stopping me from doing anything, so I’m going to just keep on pushing,’” Mr. Floyd, 45, explained.

The neuropathy kept Mr. Floyd from feeling a rock that had slipped into his boot while working. The rock caused a sore on his right foot. A doctor treated the wound, but it festered for five years. By the time Mr. Floyd learned he had diabetic ulcers, the infection had spread to his bones, leaving him no choice but to have his leg amputated from the knee down.

He lives in one of 12 states where Republicans have refused to expand Medicaid under the Affordable Care Act, citing the cost, of which states would eventually pay 10 percent. The social policy bill would close the so-called Medicaid coverage gap by offering an estimated 2.2 million low-income adults like Mr. Floyd free private insurance — but only for four years.

Unable to work after the loss of his leg, Mr. Floyd turned to singing and playing drums for a living, performing gigs with his cover band, Proximity Rule. But just as he was learning to walk with a prosthesis, he said, he noticed a lump in his neck. He waited a year, then saw a doctor, who told him he had Hodgkin’s lymphoma, a type of blood cancer.

Mr. Floyd said a social worker at North Mississippi Medical Center helped arrange for free treatments — surgery, chemotherapy and radiation. “The preventive care,” he said, “is what I couldn’t get.”

Jill Swenson and her husband were raising buffalo in upstate New York in 2009, when he had a recurrence of skin cancer. The couple had no health insurance, a factor that Ms. Swenson says contributed to her husband’s suicide. The Affordable Care Act made coverage accessible to her again in 2014, and she has had it every year since, but it was still a stretch.

She now lives in Appleton, Wis., where she is a self-employed editor and literary agent. She earned around $45,000 last year, and paid more than $300 a month for her insurance. During the pandemic, Congress has temporarily increased the premium subsidies provided under the health law — a $200-a-month discount that Ms. Swenson, 63, said has allowed her to buy birthday gifts for her niece and nephew, keep up with rising grocery costs, and pay utility bills and her mortgage.

“There’s nothing to cut,” she said. “It’s not like I’m living high on the hog.”

The temporary boost in subsidies extends up and down the income spectrum, lowering the cost of insurance for almost everyone who buys it through the Obamacare marketplaces. The social policy bill would keep it in place until the end of 2025.

The change was a response to a widespread concern that the Affordable Care Act had not, in fact, made insurance affordable enough for many Americans. More than half of people who were uninsured last year qualified for premium subsidies or Medicaid, according to an analysis from the Kaiser Family Foundation. Since the new subsidies were introduced, along with a big advertising push, an additional 2.8 million people have enrolled in coverage.

After she recovered from cancer, Shara Clark decided to become a home health aide in June as a way to give back. “When you go through a medical scare such as I did, you develop empathy for others,” she said.

Employed by an agency in Charlotte, N.C., she works 25 to 40 hours a week, helping clients get dressed, make a meal or get around.

But Ms. Clark, 41, also has two part-time jobs. “Because I’m only getting paid $10 an hour, that does not match the cost of living,” she said.

The $150 billion in the spending bill for home and community-based services has two goals. It would allow more elderly and disabled people on Medicaid to qualify for subsidized care in their homes or at community programs, helping them avoid moving to a nursing home. There are currently an estimated 800,000 people on waiting lists for these services.

But the money is also supposed to go toward raising wages for home care workers like Ms. Clark.

Home care workers make an average of under $14 an hour, or less than $30,000 a year, according to a new study from the Economic Policy Institute, a liberal group. Most of the workers are women, and many are of color.

“Wages have to go up if services are going to go up,” said Ai-jen Poo, the executive director of the National Domestic Workers Alliance, an advocacy group. “Those two goals are absolutely interdependent.”

Mariah Forster Olson’s treatment for childhood cancer left her with a range of health problems, and a long list of prescriptions. Ms. Forster Olson, 42, takes 30 prescription drugs every month and about 50 pills a day, requiring weekly trips to the pharmacy. Her Medicare coverage makes a big difference, but still leaves her with thousands of dollars in bills.

There is currently no limit on how much Medicare recipients can be expected to pay out of pocket for their drugs, a situation that leaves some who take expensive medicines with annual bills of $15,000 or more. But for the 2.5 million beneficiaries who spend more than $2,000 a year on their drugs, Medicare would pay all their costs above that amount under the bill.

The legislation would also cap out-of-pocket costs for insulin at $35 a month. That change alone could affect the more than three million Medicare beneficiaries who take the drug to manage their diabetes.

Ms. Forster Olson, of La Crosse, Wis., who qualifies for Medicare because she is disabled, said her drug costs could be as high as $7,000 next year without a change. “A cap of $2,000 would be amazing,” she said.

Anne Madison, a retired computer systems engineer in Baltimore, started losing her hearing in her 50s. Now 71, she cannot afford hearing aids, which can cost as much as $5,000. Medicare will not pay for them.

“I can’t whip out the Mastercard,” she said. “If I put that much money on it, I’ll be in trouble for the rest of my life.”

Nearly two-thirds of Americans older than 70 have hearing loss, but fewer than 20 percent of them use hearing aids, said Dr. Frank Lin, an ear, nose and throat surgeon at Johns Hopkins School of Medicine.

When Congress created Medicare in 1965, hearing aids were in their nascence, Dr. Lin said, and hearing loss was “not seen as anything remotely important.” Today medical professionals know better; beyond being “arguably the leading risk factor for dementia,” hearing loss can lead to social isolation and depression, Dr. Lin said.

But while Medicare will pay for an audiologist to diagnose it, that is where most coverage stops.

The House-passed bill would add coverage of hearing services to Medicare beginning in 2023. Audiology services, including counseling for hearing aids, would be reimbursed, and the devices themselves would be covered for people with “moderately severe, severe or profound hearing loss.”

Ms. Madison, of Baltimore, is addressing her hearing problem another way for now: She enrolled in a study at Johns Hopkins that evaluated over-the-counter hearing amplifiers that cost about $150. She is now able to attend meetings and church services, and has pleased neighbors by turning down the volume on her television.

And, she said, “it was amazing to me to be able to hear my little grandchildren.”