Universal healthcare is such a moral sham, and what we have now in the US is also pretty bad. There is a better way.
If you're trying to get enough hospitals to do this that people don't just go to a cheaper hospital, then almost all hospitals are going to have to raise their prices... and that's effectively the same as a tax. They're "robbing" every patient the same way.
The hospital can turn of course offer to foot the bill, but can do so only after you’ve exhausted other options. But none of this is forced.
1) More expensive healthcare, to cover for these cases directly in a fund or via third party insurance.
2) Healthcare scarcity.
You can't draw blood from a stone. Healthcare investors/workers at large don't get in to become a charity so your "solution" comes with explicit downsides you should mention.
No solution is perfect, but that’s a world that’s more personal for those helping and being helped. It also is more economically efficient. I’m not saying this is the only solution, but one that starts with robbing your neighbor is the moral low route and we should and can do better.
Taxing everyone to hell to cover every health expense is immoral.