One downside of this though, is that ICU beds are full. I'll also hope I don't need one. I very rarely have before, so I'll probably be ok.
Only if they, y'know, survive, and only after being a nuisance to the healthcare system that the rest of rely on
You've got a point that verifiable natural immunity as of _now_ should probably be acceptable in lieu of vaccination (though efficacy against variants is worth considering) for mandates.
Offering 'natural immunity' as an option to the currently un-vaccinated un-infected population is just crazy. People will assume the risk, get sick, got to hospital, and die. Just ignoring those 'others' isn't an option for public-health since the downside costs of the risk they assume are borne by everyone in the form of healthcare burden. 'They' still expect to be a priority when they get sick...
If you don't want to catch the virus, get the vaccine. Don't worry about others.
They will get the virus and then get natural immunity. People will have adverse reactions to both covid and the vaccine. Let them weigh the risk to which one.