I see where you're coming from and I don't entirely disagree with your conclusion. The post you linked, makes a strong case for not jumping to conclusions based on grey matter studies, which I think is sound advice.
That said, I maintain that given the option to get COVID or not get COVID, I would greatly prefer the later. You similarly won't find me gaming long hours, and I make sure to get plenty of sleep. In other words, taking precautions that avoid injury is generally a good idea. And FWIW, I don't find the controls in that study to be terrible? I'm not exactly an authority of statistical analysis though, so I'll trust that this is outside my scope of understanding.
Where it sounds like we disagree most is whether or not the risk of COVID causing injury is worth something as small (or large) as asking for proof of vaccination.
It just so happens that I think asking for proof of vaccination is a relatively minor thing given the possibilities of COVID.