Re: I've had a vaccine passport for decades
Yes, and it does its job very well. Despite just being a bit of yellow folded paper. No photos, no significant anti-forgery features, no centralised control over issuing (high street pharmacies have stacks of them for issuing when they give the vaccination), etc.
We do not need to sacrifice privacy when proving we have received the vaccination. For international purposes they will need a name on them (like the yellow fever card), to be matched against a (separate) passport, for domestic use the current cardboard card should be perfectly acceptable, with no identity information at all. If I am going to a play or to a sports match I have no need to identify myself.