Apart from the medical/pandemic aspect, there might be an interesting comparison with the implicit agreement to be searched at the turnstiles. In Germany (where fans tend to be more boisterous than at Firhill, even at "lower" grades of football) you either agree to this or don't gain entry. AFAIK nobody has ever objected. (The police, who attend major matches, are armed, btw.)
In much the same way as we generally carry fistfuls of cards - credit, driving licence, season ticket, etc. - but express shock and horror when someone suggests ID cards, is objecting to a vaccine record not the same emotive rather than rational response?
(On a personal note, if I'm travelling to a country with tropical diseases, I ensure that I have a record of my vaccines with me, just in case it's useful for medical reference. That might be a potential employer, rather than a medical practitioner.)