What sort of things will health checks be used for in the future? Once Covid is gone, I don't think people will really care about your health information beyond entities that already care (health insurance providers denying coverage because of preexisting conditions, etc.) Certainly, The Mets won't care if you got your flu shot this year, and your local bar probably won't care that you didn't get your annual physical. If unrelated companies start caring about your health information after the pandemic, we should tell them to fuck off.
Covid is a public health crisis -- the single most dangerous existential threat we as a society are currently facing -- and we have great tools to kill it dead. Letting it spread right now is just stupid, because we can end this plague once and for all. While we're working on it, I think it's totally fair to let someone check your vaccination status before you pack yourself into close quarters with them. It's not a matter of the government or the corporate overlords wanting your health information. As the person standing next to you in a crowded bar, I don't want you there if you're not vaccinated. If indoor businesses won't check customers, then I won't go to those indoor businesses, and if enough people think like me, those businesses are gone forever.
Get vaccinated or stay home. You have a choice!
(The real threat during crises like these is things like "we shouldn't have a free press, because nutjobs are spreading misinformation". That is an exceedingly dangerous line of thinking and we should push back against that hard, because the cure is worse than the disease. That's not the case with Covid -- the cure is much better than the disease.)