But the party's emerging healthcare proposals would shift even more costs to patients, feeding the very problem GOP politicians complained about under Obamacare.
And their solutions could hit not only Americans who have Obamacare health plans, but also tens of millions more who rely on employer coverage or on government health plans such as
Many in the GOP also want poor people who rely on Medicaid to face more co-payments and higher premiums, citing the need for patients to have "skin in the game."
Developing House Republican plans to replace Obamacare would scale back government insurance subsidies for millions of low- and moderate-income Americans who rely on the aid to buy coverage.
To fund a healthcare overhaul, Republicans are exploring ways to scale back tax breaks on health insurance, a move that could mean a tax hike for people who get coverage through an employer.
Meanwhile, other GOP plans to overhaul Medicare — which Ryan and Price have championed — would provide seniors with vouchers to shop for commercial health plans, an approach that independent analyses suggest could leave many patients paying more.
Those are politically risky ideas, said Robert Blendon, an authority on public attitudes about healthcare at
The GOP proposals — many bedrock conservative healthcare ideas — also could prove a major obstacle as Republicans labor to convince increasingly skeptical Americans that they have a better alternative to Obamacare.
Republican leaders like Ryan contend their ideas will transform healthcare markets across the country as government regulations are pared back, driving down costs.
"We believe in a patient-centered system, where individuals have the freedom to buy what they want and not what the government makes them buy," Ryan told reporters at the Capitol recently. "It's really, really important to have choice and competition in healthcare because choice and competition lowers cost and increases quality."
GOP lawmakers have tapped into fertile ground with their relentless attacks on rising costs. Though insurance premiums are generally increasing more slowly than they had in the past, many consumers are still seeing steep increases even as their deductibles also balloon.
The increases have been particularly dramatic on the insurance marketplaces created by the healthcare law, which serve only about 11 million people but have become the law's most visible program.
This year, the average annual deductible for a silver plan for a single adult on the marketplaces hit $3,572, according to a survey by HealthPocket, an online insurance shopping tool.
Most experts believe high healthcare costs are being driven primarily by the high price of medical services in the U.S.
The average cost of a hip replacement, for example, is nearly twice that in Switzerland or Britain. Many brand-name pharmaceuticals cost two or three times what they do in other industrialized countries.
But most GOP proposals focus on getting patients to pay more out of their own pocket for medical care as a way to control rising premiums.
A favorite Republican strategy is to increase use of high-deductible health plans, coupled with tax-free health savings accounts, or HSAs, which consumers can use to set aside money for medical expenses.
"HSAs tied to high-deductible health plans are popular tools that lower costs and empower individuals and families," House Republicans said in their "Better Way" blueprint for replacing Obamacare.
Under current law, high-deductible plans can require single people to pay as much as $6,550 before their insurance kicks in. Families can face deductibles as high as $13,100.
Consumers who have qualifying high-deductible plans can put aside as much as $3,400-a-year in a health savings account, or $6,750 of they have a family health plan. Many Republicans favor allowing Americans to put away even more in these accounts.
Republicans similarly favor more cost-sharing in government safety net programs, such as Medicaid.
The federal government historically has limited how much state Medicaid programs can charge poor children, pregnant women, and elderly and disabled patients for medical care.
Trump has said he has no plans to alter Medicare, but senior officials in the Trump administration, including the vice president, helped pioneer broader use of premiums and co-pays for Medicaid patients in Indiana. Many in the GOP would like to see this approach broadened nationally, including the woman Trump has tapped to oversee the federal Medicaid program. Seema Verma also worked in Indiana.
Price, Trump's Health secretary, has also called for changes to the Medicare program that could hit seniors with higher medical bills through a practice called balance billing, which allows physicians to collect a surcharge from patients beyond Medicare's fees. Balance billing is currently prohibited.
Even as Republicans envision patients paying a larger share of their medical bills, many GOP leaders are also calling for cuts in federal aid to help Americans pay their insurance bills.
Among other things, congressional Republicans are discussing plans to scale back subsidies made available by Obamacare that help about 8 million Americans buy health coverage on state insurance marketplaces created by the law.
These subsidies are linked to consumers' incomes, with much more aid available to poor people. If the system is changed, millions could see their insurance premiums jump.
The effect of such cuts would be limited to the small fraction of Americans who rely on the marketplaces.
But Republicans are also eyeing changes that could affect health insurance costs for a growing share of the roughly 150 million people who currently get coverage through an employer.
Under plans being discussed by House GOP leaders, the federal government would scale back the tax break that employers and workers get on health benefits. For decades, health benefits have not been subject to income taxes or payroll taxes.
Changing that could make health insurance more expensive for many workers and families, and disrupt a highly popular system. In one recent survey, more than 8 in 10 Americans rated their employer-based coverage as excellent or good.
The GOP plans to shift costs to patients with high medical bills and less government aid are already generating skepticism among many experts, who question whether most Americans will be able to put aside more money to pay for their healthcare.
Nearly 70% of Americans in one recent survey reported having less than $1,000 in savings.
At the same time, it's unclear that loosening government regulation of health insurance will produce the substantially cheaper health insurance that Republicans have promised.
“There is this assumption that premiums will fall a lot,” said Len Nichols, a health economist who heads the Center for Health Policy Research and Ethics at
Most immediately, GOP leaders face a more serious political problem of convincing Americans of the merits of their strategy. "Nobody ever says they want to spend more on their healthcare," Harvard University's Blendon said.