Why? No evidence is needed that vaccine mandates are justified, as if evidence were required it'd fail immediately given the low and falling efficacy, that vaccines are individual protection so it doesn't matter what other people do, and given that hospital capacity is the problem of the hospital providers to solve and not their customers (this is fairly basic).
But hey, if you disagree with that last bit and think hospital staffing is everyone's problem, be aware that this sword cuts in both directions. It actually makes it super easy to argue for banning vaccines. Here's the argument they'll use:
1. Vaccines aren't entirely safe. They can cause blood clots, strokes, heart attacks and many other kinds of injury.
2. The injured end up in hospital, consuming medical system capacity which takes it away from the people who opted out and didn't take the vaccines. Possibly for life, in some cases.
3. This is selfish because many of the people who took vaccines and got injured are young and weren't at any meaningful risk of COVID. They did it to themselves despite knowing there were risks, because they were incorrectly scared, or authoritarian, or whatever, but now society has to take care of them anyway.
4. And in addition the unvaccinated are now almost all recovered, and they have natural immunity for life. The vaccinated on the other hand, are now dependent on endless booster shots which not only injure new people on every round but are also are extremely expensive and taxing on the healthcare provisioning system, meaning less resources available for everyone else. Proof: the UK has just started cancelling operations to free up healthcare resources for booster shots!
Thus, the vaccinated are a drain on the healthcare system and should have their freedom to choose endless boosters removed - or at the very least, they should be forced to pay for them themselves (and any subsequent healthcare issues they experience).
You may not like the above argument (I don't!) but it is not fundamentally different to the ones being deployed in this thread. And if the unvaccinated become the majority, as may easily happen given that vaccinated people will constantly be joining them as their status expires, then nothing in theory stops such an argument being made (although in reality the public health establishment would never go along with it).