You've hit an important central issue: that Governments (and/or HC systems) aren't ready to take this step. As such, I suspect it might be a logically easy concept, but as a political step, maybe not so much. :)
I mean, in the absence of a few thousand years worth of medical ethics, sure, that's an obvious step.
But it's very much not how things are done. The only place where anything even vaguely like this happens is with organ transplants. In general, hospital triage is based on need; if the unvaccinated person with covid is sicker than the blameless road traffic accident victim, and there's one ICU bed, well, it doesn't matter that they brought it on themselves; the covid patient gets it, and good luck to the road traffic accident victim.
This isn't _just_ a political issue (though it would also be a very difficult political issue); it is how the whole institution works.
Triage based on vaccination status would be a very extreme move, much more so than mandatory vaccinations to work or than unvaccinated-only lockdowns.
It's political suicide to do so. Takes courage to do the right thing.
You know that the vaccine has been available for, like, nine months, right? Anyone who wanted to get it got it months ago, and has been rather safe. ICUs being full points to a problem with the medical system (and, at least in America, the medical system is well known to be broken), and if you don't want to fix that, there's a really easy political fix (which is to de-prioritize unvaccinated COVID patients, given that - again - there's a vaccine available, so if you willingly chose not to get it, then your medical care can wait until those who have unpreventable issues are helped).
We're well past the point where you can make the argument that these restrictions are for the greater good, because we've been in a state where one's health is almost entirely one's personal responsibility (you either take the vaccine and you're safe, or you don't and you're not) for months now.
If other people want to remain unvaccinated and that leads to them dying ("loss of human life on a massive scale"), they can do that.