Done properly, there is no surveillance, no record of where your code is being scanned. This includes by the scanning app. The SMART Health Card format at least, contains everything needed in the code, and little else. The app just needs the issuer's public keys to verify the signature.

That the anti-everything crowd will pout and stay home is precisely the point.

Hasn't everyone in the modern world lived under a "papers, please" surveillance type system for multiple decades now?

This is mostly only true with international travel.