Bars already check licenses before serving alcohol.
Schools already check vaccination status for other diseases during enrollment.
Workplaces already check immigration status for all employees.
There will always be non-compliance; there are always a few bars that don't check IDs or accept obvious fakes. But most won't risk losing their liquor licenses.
There are a lot of "what if"s in the world, and in public policy in particular. What if no one pays their taxes? What if a state legislature overrides the popular vote of their electorate in a Presidential election? What if the President orders a nuclear first strike and the military doesn't follow the order?
While all of these questions are interesting, for the most part we can get by without having any answers to them. Why borrow trouble worrying about something that may never happen?
What if workplaces mandate vaccines, what if schools, if bars and ballparks and theaters, and we still don't reach the 90-95% thresholds needed to contain the more virulent viruses?
Well, something else will happen. Maybe we give up. Maybe we try other coercions. Maybe the combination of vaccination rates and post-infection immunity will be high enough that the entire question is moot.
Or we don't do any of these extreme, totalitarian measures because it doesn't matter anyway?
Because doing so is a one way road to more extreme, totalitarian measures next time there's a disagreement about public policy (and next time, perhaps you find yourself on the "wrong side").
Because covid won't be a thing forever, and sacrificing core values of your nation for temporary gain is foolhardy.
Because, again, the people you're trying to protect with these extreme, totalitarian measures don't want your protection and statistically will turn out just fine anyway.
Not to mention turning every business into some sort of citizen police is a very weird concept in the US.