It's a horrible idea! Even assuming we were confident in the data around immunity.
For one, it may lead to people deliberately getting infected out of desperation.
As a tech sector executive, I'm also recognising another familiar pattern: if it's new and "sexy" in technology, slap it onto everything you can because ... reasons. Face recognition is utterly unneccessary for this purpose. A piece of paper or a digitally signed app in combination with any piece of photo id suffices entirely and is a lot less intrustive.
This reminds me of that time a few years back when some idiot colleague of mine wanted to pitch the idea of tracking pilots using blockchain to a major airline client: first of all, it's not exactly as though there are a) multiple parties involved in a situation where no inherent trust can be built. It's also not as though b) it's at all likely that a pilot's leg would be silently switched with a shorter one in mid flight and noone was likely to notice. Pilots are physical being, not data! Long story short, this was obviously a fucking stupid idea that solved a problem long solved in a new and entirely unneccessary manner - but "blockchain" was sexy and hence had to be pushed. Happy to report we never did pitch this thanks to my boss being eminently sensible and listening to reason despite his initial enthusiasm.
Sometimes I despair of my industry. Let tech solve new problems or improve on things. That's what we're good at.
Also, we generally suck at epidemiology. It's not our field!