He vaccine while not designed for the current variant does provide some protection. If it didn't we'd see hospitals filling up and infection surging. That's not happening because even if people vaccinated catch it because the antibodies aren't effective, the T-cells are and it's basically just a cold. So this it's not a big threat to the healthcare system. But that's only true because so many people are vaccinated.
This is a specific logical fallacy called
affirming the consequent. You are saying "If the vaccines were working, the hospitals would not be overwhelmed. The hospitals are not overwhelmed, so the vaccines are working."
The hospitals could be in a not-busy state for many other reasons. Milder variant (almost certainly true), and summer season (seems true, since Summer 2020 was also quiet, before vaccines were available), to name two.
The all-time peak of Covid hospitalizations in both the U.S. and Canada came after the vaccines were administered, not before. So no, I don't buy that they're the only thing standing between us and a collapsed healthcare system. Not at all, actually.