Why do you think that R would be below 1 with a 100% vaccination rate? In countries with 80-90% vaccination rate we still see massive spread. Handwaving about "nonlinear effects" doesn't cut it. Vaccine mandates are a very heavy tool, that needs very heavy justification.

There is no magical law of physics that ensures that a 100% vaccination rate gets R<1. It all depends on how transmissible Omicron is in vaccinated people.
The numbers are currently still in flux, but a reasonable current estimate is that two vaccinations are approximately 70% effective against hospitalisation with Omicron.
We also know that it is (much) less effective against transmission than it is against hospitalisation.
Even if it were 70% effective against transmission, that might still not be enough to get R<1 given the ease at which Omicron spreads.

Compare, say, with the measles, which has a natural R of about 10. With a 70% effective vaccine, it would still spread. Luckily we have a vaccine that is 99% effective against transmission, so it can be stopped. COVID vaccines are not nearly as good.

To give some indication from my country (the Netherlands) where 86.3% has been fully vaccinated, of the positive corona tests, approximately 66% fully vaccinated. We can see that this 66% is less than 86.3%, indicating that the vaccination is somewhat effective at stopping transmission, but that's still a heck of a lot of vaccinated people who can catch and transmit covid.

> Even if it were true that R would still be > 1 with a 100% vaccination rate, as long as the R rate is lower with the vaccines than it is without the vaccines, the vaccines are preventing (some) transmission.

Of course, but in the long run it would make little difference, because as long as R>1, then more and more people will continue to catch the disease, just over a longer period of time. Smearing out infections can be important if there is a shortage of hospital beds, but this can hardly be called "stopping transmission".

> The more contagious the virus, the higher your vaccination rate needs to be, for reasons that I hope are obvious.

Absolutely correct! And at a certain point, unless the vaccine is 100% effective against transmission, the vaccination rate would need to go above 100%, which is clearly not possible.

Secondly, many lifestyle choices affect the rate of transmission. The singular focus on the forcing people to inject something into their body against their will is unethical (note: I have personally been vaccinated). It is unethical not just because it violates bodily autonomy, but also because it is an easy way to abdicate from your responsibility to take care of your other lifestyle choices that affect the spread.

> We really, really need to starve the virus of hosts, particularly so we don't breed ever more variants.

Not going to happen. Even at a 100% vaccination rate it would still have many human hosts as the data I quote above show, but also because many wild animals now have COVID too. Unless they develop a new vaccine that is much more effective than the ones we currently have, there is no stopping it any more. We need to learn to live with it, like we do live with the flu. We are lucky that Omicron is so much milder than Delta. If you are fully vaccinated then the chances that you have serious illness are very very low. It for sure is terrible for old people and those with existing health problems, but we cannot realistically do anything about that, except for instituting Chinese level control. The response against COVID is at this point completely irrational and doing more damage than COVID itself. Not just economically (which directly translates into future deaths), but also because it is tearing societies apart.