You could. However, this would require the prover to know how to evaluate the property -- in other words, the test for CSAM would have to be publicly-known, or

You could. However, this would require the prover to know how to evaluate the property -- in other words, the test for CSAM would have to be publicly-known, or at least known to the client. This is not what providers and law enforcement want, since this would amount to giving away an algorithm that smart criminals could use to pre-test their content to see if it'll trip CSAM filters. To date, organizations have been extremely cagey about giving out content hashes for CSAM as well as even the algorithms for computing them, like PhotoDNA (which recently leaked.) The more recent proposals use two-party computation so that the client doesn't know whether any given piece of content matches. You can attach ZK proofs to the client's piece of this computation, all at fairly large (computational) cost.