Again, it is not the government. It is private companies. This isn't a "papers please" situation. People volunteer to get extensive background checks to qualify for TSA pre-check and avoid the longer lines/shoe removal hassle at the airport. It's not required, but people give up personal information for a perceived benefit. Is that the airlines doing government bidding?
Schools require up to date vaccinations to re-enroll. Kids on sports teams have to get a physical and release that information to play HS football. We voluntarily provide health information all of the time because the benefits outweigh the perceived risks/concerns. Why is this now, such a big deal?
I'm sorry, but this just seems like such a weird hill to die on. If you don't want the vaccine, cool. If you want to get the vaccine but don't want anyone to know, cool. If I choose to do both, why do you care?
I honestly can't imagine what my most despised politician would do with my vaccine status information that would cause me harm.