Specifically I'm wondering what the maximum number of cases in a short period is likely. I ask because I was shocked to hear that the number of ICU beds in a major city of 1M people in Canada was only a few hundred (or maybe that was total hospital beds? I can't remember). So it seems like you need to keep your "hospitalizations per population" number down to less than the ratio of "hospital beds per population".
Certainly I was wrong in saying "everyone will get it" in a few months because of the logistic growth curve, but it looks like a significant portion of the population could still be infected within the same period, even if you remove the 6% (Canada) to 20% (USA) people who have already been infected? And even if everyone is fully vaccinated, can the number of them that end up in hospital come close to the total capacity? It seems like we're already getting close to capacity.
As for your original question – I've been wondering about the same thing.
My personal back-of-the-envelope calculation was more along the lines of "If the vaccine is 90 % effective (the numbers given for Germany during the Delta wave and before the start of the third round of vaccinations were closer to 90 than to 95 %) in preventing hospitalisation and 75 % (for example) of the population are vaccinated, it means that case numbers in hospitals are 75 % x 1 + 25 % x 10 = 3.25 times higher than they would be with full vaccinations.", though I suppose there are enough holes that could be poked into that calculation, too.
A three-fold reduction in case numbers is nothing to be sneezed at, but on the other hand as long as the spread is still behaving mostly exponentially instead of turning into logistic growth and slowing down again that's only one-and-a-half doubling periods, so not that much of a buffer, either. Of course the vaccines are also somewhat (if not as well as was originally hoped) preventing some spread, so with full vaccination growth rates themselves would also be lower and hospital cases be reduced by more than threefold, but whether that's then enough, who knows?
In the end I decided there's no point in worrying myself further here, because there are too many unknown factors in getting a reasonable result that I can't easily answer and ultimately I have no desire in becoming a full-time epidemiologist.
As people become infected and gain immunity (in reality not always perfectly and not necessarily until the end of times, but on average they'll definitively do, and right now we're only talking about a simplified thought model anyway), they drop out of the pool of people available for further infections and therefore slow down the growth rate.
While still only a simplified approximation, logistic growth is therefore a better model than naive unchecked exponential growth, and the maximum case rate will be somewhat lower than "half the population gets infected in the last doubling step".