Single-mindedness is not a rationale for imposing medical treatment on people.
I'll circle back to my driver's license analogy - the bouncer does not need to know that I was docked 3 points from my driving test for failing to signal - he only needs to know that some granting authority, with expertise in the realm being licensed, granted me the license. (or that I'm old enough to enter the bar, etc.)
When 1500 Americans are dying every day of a preventable disease, it is absolutely reasonable to take feasibility of enforcement into account. And enforcement has to be done where the greatest risk of spreading occurs. People might not like getting carded at a restaurant, but there is a point where lots of people are dying and it makes sense, right? We card for alcohol, we card for driving, and those we do every day.
I would think someone who tested positive should be able to get a ca state qr code. It seems the science supports that. Having each validator check that the test is the right test, and the doctor and hospital exist, seems not feasible, but getting a state qr code - then using that at the point of enforcement - seems practical.
But I must remind that, at least in this particular conversation, we're talking about people who've just had the illness in question. We don't know how many of the 1,500/day are people who have previously contracted the illness or else contracted the illness from someone else who previously caught the illness, but in all likelihood the proportion of such cases is likely very low. So referencing the 1,500/day likely has very little relevance to a discussion of natural immunity.