I think I'm failing to make this clear enough. I'll try again.
The set of unvaccinated people is made up of those with a prior infection (call them R for recovered) and without (call them N for naive). And for simplicity, we'll keep the vaccinated variable to a binary, yes or no. So you have vaccinated with a prior infection (VR) and vaccinated without (VN).
When you say unvaccinated you are talking about R + N. We have data from the CDC that says R and VN were at roughly equivalent risk of hospitalizatiom during delta, and VR was a little bit better. N was by far the worst, and is who you actually mean when you talk about the people disproportionately filling up the hospitals. Because that's what the CDC is saying, that members of R, VR, and VN were all showing up in the hospitals at a similar much lower rate, and N was in a different universe.
You wouldn't know this unless you specifically asked yourself the question and went looking for the answer, because so far it has gotten very little air time. So it's not surprising that you and lots of other people are not aware. I'm only aware because I happen to be a member of R, and so this is important to me. It sucks living in a world where lots of people think its fine to lump me in with N, and advocate for taking away my livelihood and freedom of movement.
Saying "the unvaccinated are clogging up the hospitals" is like saying "I don't like Mexican food" when really you just don't like spicy peppers. You may avoid some burn that way, but now you're missing out on the rest of the cuisine, and you've failed to equip yourself with the knowledge to avoid the Indian or Thai dishes that will destroy you (because you think it's just the Mexican part that matters, not the spicy ingredients).
Does that make sense?