Yeah this sure is a common talking point that comes up, but unfortunately for those who enjoy enforcing these things, it is not backed up by any data.

Daily deaths (which is the best proxy we have for hospitalization) is nowhere near any of the peaks we have had in the last year and a half.

If somehow deaths are down but hospital capacity is also down, that is the fault of the hospitals, not the vaccinated. https://www.worldometers.info/coronavirus/country/us/