but that also comes with the issue of trusting the miner whether they wouldn't take the puzzle reward themselves or not
Well, it is also possible to share some data, which is needed to mine a block (like transaction ID, transaction Segwit ID, etc.), without sharing transaction data. Then, it would be sufficient to mine a block, but the pool wouldn't know, if it is valid or not.
And also, if you want to prove, that you know the public key, but you don't want to reveal it, then you can share for example SHA-256 of that key, and then everyone can validate, that RIPEMD-160 of it is equal to the address, used in the puzzle.
For example:
This is the public key, which you want to keep secret: 04678afdb0fe5548271967f1a67130b7105cd6a828e03909a67962e0ea1f61deb649f6bc3f4cef3
8c4f35504e51ec112de5c384df7ba0b8d578a4c702b6bf11d5f
This is what you can share to prove, that you know the public key: 261c1eb21fc4708c6acbe1cfc6d4565652e9e768b620782898936b93000a6c02
This is the hash used in the address: 62e907b15cbf27d5425399ebf6f0fb50ebb88f18