The government shouldn't know if/how many times I use my ID—you would be essentially building a country-wide blackmail database since it's a near direct proxy for porn usage. And it doesn't even matter if it's true, people will assume it anyway.
Your system effectively collects exactly the data ZKP is intended to protect.
Which is a long way of saying "ZKP" isn't an answer to this problem because you can't actually have zero knowledge in a system where people have little incentive to keep their key a secret.