Besides, this will require a complete rewrite of the bitcoin protocol, which is so used to the methodology: transactions --> blocks.
No rewrite needed. Including or excluding a particular transaction is a mempool-level decision, so each node can have its own rules for that. It would be harder if you want to distribute newly generated coins, based on that work, but probably also possible.
Average people would use smartphone or computer CPU to perform PoW, while attacker could just use multiple GPU or even QC itself to break PoW requirement easily.
True, it depends what will be broken, and how it will be broken. Because I can imagine a situation, where some attacker does not know the private key, but can produce a valid signature (for example because of public key recovery). Also note that if SHA-256 is broken, then ECDSA in the current form will also be, because you can easily generate a signature with some random hash, and then if SHA-256 is broken on preimage level, then you could just construct a transaction matching that hash.
The PoW for the tx does not necessarily enter the blockchain
If it is only to include or exclude some transaction, then yes, it could be a commitment, based on some new algorithm, and then it can be processed only on mempool level, and then dropped if some transaction will be deeply confirmed (as long as this new algorithm is not directly supported by consensus rules).
it could be a segwit bit of data but actually is easier just reuse the actual signature and keep resigning such that the signature is the tx PoW nonce
You don't need witness space for that. Using a commitment, that will be processed only by upgraded nodes, is cheaper, and if everything is about mempool rules, then that kind of data can be completely separated from the current consensus. So, all that is needed, is just tweaking R-value of the signature, so it will commit to some new hash function, and a new signature scheme, intended to be used in the future. Then, current nodes will just see some random R-value, but upgraded nodes will also process a commitment from the new version.
If nothing else, it would be interesting to include a PoW check on itself within the script
You don't need that, because there are no consensus rules, that are related to including transactions into blocks. You have to include the coinbase transaction, but everything else is up to the node, and you can touch all of those settings, without touching consensus.