Long answer: you'd probably come up with a protocol around "ok, we have this ventilator available, give it to the unvaxed patient, but if we have other patients who end up needing one and they haven't improved in X hours/days, we will reallocate" and similar for all the various ways people can present, but "conveniently"... if the health system is that overloaded, you're gonna have ERs with a lot of patients all arriving at the same time.
Hell, even pre-COVID, I got stuck waiting in an ER in December for days because the hospital was full once.
(To your edit: has either smoking or obesity caused hospitals to fill to capacity and have to turn away patients? You'll notice here I'm talking about scarce medical resource allocation prioritization where they're actively competing with those more considerate of the rest of us, not something like insurance payments or coverage.)
If two do arrive at the same time, an unvaccinated person with far worse sickness faintly mumbling about ivermectin who waited until the last minute to come in, and a vaccinated person who is suffering but nowhere near the danger zone - do you put the unvaccinated person on the sidewalk outside?
What about smokers or fat people?