To answer some of the sentiments here, I can't foresee a way this could ever replace either proof of work or proof of stake.  You have to have miners of some form to run the chain and only then would transactions work.  So this isn't a replacement only an addition.

One way to look at it is that it can reward people fkr sending transactions, like participating in a lottery.  Sending a transaction would be like buying a lottery ticket.

The mining pool wouldn't be able to make free attempts at this, at least not at a reasonable level because only after the blockhash is made (after hashing together all the block data) will you know if anything is a winner.  So creating a transaction id is just a ticket  and any winners will not be able to be determined until all of the block has been created and preferably even if that block gets accepted by the rest of the network, so perhaps even include the hash of the next block.  So that would require hashing the transaction with the blockhash of that block and a future block.  Perhaps even go 10 blocks deep before knowing if a transaction is a winner.

I think it would be cool feature of bitcoin, but ultimately it could also be powerful in defi, where you can tune the difficulty so every 1 million transactions or so new coins are minted and added to liquidity, or a big airdrop of coins is dropped on every holder or something.  And these airdrops would scale in frequency based on how popular the network became and how high the transaction volume becomes.  If people aren't using it, few coins are minted, but as people use it a  lot, more tokens would be minted.