Let's do some napkin math for the US and see how this works out.
Assumptions (all from worldometer and ourworldindata):
"Most estimates have put the number of undetected cases at several multiples of detected cases".
Roughly 43 million confirmed cases recovered so 120m if we use the smallest multiple for "several".
330m total population.
74.5% with at least one dose (245m)
The set of people who are likely to have an immune system primed enough to stave off severe illness (all else equal, many will have preexisting conditions that make them at risk anyway) should be (#dosed + #recovered) - #(dosed AND recovered).
(#dosed + #recovered) is 365m (using 3x cases underestimate factor).
#(dosed AND recovered) has an upper bound of 120m if all recovered got dosed, and a lower bound of 35m (not 0 because the whole equation has to add up to total population).
So (if my math is right, no promises) the US has between 245m and 330m immune primed people today. If its anywhere near the 330m end of the spectrum today, it will be there in the next 2 weeks/30 days as the confirmed case counts go over a million per day (thats only counting confirmed cases).
The best case for mandates out of this: if we had the max of 85m seronaive and you could jab them all at gunpoint tomorrow, a huge chunk of those people will have gotten omicron before the shots take effect. You might prevent some hospitalizations before omicron burns itself out. Good luck ever getting these people to vote for you again.
In the minimum seronaive case, you prevent very few hospital visits, you create a bunch of adverse effects, you waste doses that could have gone to undeveloped countries, and you still really hamstring yourself in future elections.
Everyone's immune system is going to be primed anyway. The mandates are only an expensive footgun at this point.