The UK probably has a large chunk of the benefit it could ever possibly get in terms of stopping the hospitals from being overwhelmed - vaccination levels are very high in the age groups that're most at risk here.

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/

You seem to be missing another option: that treatments get better so that hospitalization leads to fewer deaths than it did earlier. Which seems obviously true.

Why would deaths be the best proxy for hospitalization? Why don't you have actual hospitalization numbers?

Did you read the linked article? The governor of one of the largest US states asked hospitals to delay elective procedures to ensure hospital capacity.