Over a decade ago, we had exactly this debate.
The House, being the House, said let's set up a national marketplace with financial support to help people without offers of health care coverage at work buy coverage, and we'll make Medicaid a universal coverage program for the poor.
The Senate, being the Senate, said let's set up state-based and state-run marketplaces (with a federal backstop for states not interested in operating and setting policy for their own marketplaces) with financial support to help people without offers of health care coverage at work buy coverage, and we'll make Medicaid a universal coverage program for the poor.
("Conservative" Republicans largely boycotted the conversation).
The Senate won. And the SCOTUS subsequently said fine, but the Medicaid part requires states to opt in to making Medicaid universally available to poor people.
Red ("conservative"?) states largely said 'we're not interested in operating our own state marketplaces, we'll default to the federally-operated option.' The bluest states said 'we'd rather operate our own marketplaces at the state level ourselves, no thank you, federal government.' What a twist! (In Mississippi, the GOP governor and the GOP insurance commissioner feuded over this very question, before the governor won and the state defaulted to the federally-run option). So most red states ended up with federally-operated marketplaces for insurance and many blue states ended up with state-run marketplaces.
Meanwhile, virtually every blue state (and most red states) accepted the option to accept the expansion of Medicaid into a program of universal coverage for the poor. But some of the reddest states held out. But proponents in some deep-red states decided to ask voters at the ballot box and it turns out deep-red voters in Idaho, Missouri, Nebraska, Oklahoma, Utah, and South Dakota all thought Medicaid should be expanded to cover all poor people.
Maybe the moral is that "a conservative approach" to this question is completely meaningless at this point. And we should stop hiding behind labels and abstractions and just do what we think makes sense.
Certainly that approach has improved the quality of life for millions of people over the last 12 years or so with respect to their health care.