A private health system can work. There's nothing inherently bad about it. A private system with some subsidies for poor people can work fine.
The problem is that in our health system, the incentives are misaligned. There is no market for insurance policies outside the ACA, most people have no power to choose their insurance provider.
One easy fix is to prohibit employer-sponsored insurance entirely. At most, allow employers to provide vouchers that can be used only for healthcare purchase.
To give my personal opinion, if given only choices on the poles (fully private vs single-payer), I’d vote for single-payer. And I’d actually do it because I think the ROI would be enormous, though difficult to quantify. I do think it’s tricky (at best) to have health care fully private because the profit motivation can be cut throat. Maybe you’re right though it could be private but with heavy regulation (sort of sounds like what we have today though for working-age people). And you didn’t say regulate it but I’d have a hard time agreeing private could work without regulation.
The belief, and this is very apparent at the moment with all the stories about DOGE and the Trump admin, is that the government wastes money and the private sector can do it better. In cases where there's no return on investment to be had, Republicans would want that "free," government-provided service/product to be cut entirely (again, see recent DOGE actions). Btw, when I say ROI, in the eyes of Republicans, the more nebulous the ROI the more likely they are to say "kill it" because this really is about dollars and cents (with some culture war issues thrown in, but that's a very small part of the larger dollars and cents issue that almost all Republicans care about).
And to put a finer point on it, the Republican Party believes a significant percentage of the US population doesn't work hard enough (or at all) and that therefore those people don't deserve to get anything for free (i.e., I don't want to pay taxes or have the government issue debt for the sake of people who haven't earned it). That's why you'll have a hard time arguing that very wealthy people shouldn't take advantage of the government because in the eyes of Republicans those people have, literally, earned whatever they get, including "free" things from the government (this mostly takes the form of pro-business regulations/subsidies).
More specific to single-payer health care, if the US offered it, Republicans believe it would take away an incentive for people to work because for working-age people that's how you traditionally get health insurance. To repeat from above, no work = nothing free. That is a feature of the system in the eyes of Republicans, not a bug.
For those wanting to understand more, ask ChatGPT/Google/etc. for Republican proposals for how to manage Social Security, Health Care in general, Medicare specifically (see "Medicare Advantage"), Medicaid, food stamps, Social Security Disability, etc.). If you really want to understand it fully read it straight from the Republicans' own websites. Visit the Heritage Foundation, or Americans for Tax Reform). Every one of these topics is discussed in detail. Particularly relevant at the moment given this administration is doing much more to make all of this a reality. Prior Republican admins did some of it as well, of course, but not as aggressively as this one.
I say this on a lot of my (mostly long-winded posts--sorry!): I'm attempting to be factual here. If you're reading a politically-biased angle please reply and say so, and if you can tell me which words in particular are triggering you, I'd genuinely like to know because my intent is to communicate the rough facts on the ground.
edit: changed one incorrect word