Suppose one of the big mining pools, that holds e.g. 20% of the mining power, creates a wallet software that, by design, sends all transactions only to pool members - and does not relay it to nodes outside the ppol. Will this give them an advantage in getting mining fees? What keeps the big mining pools from doing this?
What is the point? The service you get as a user is worse, with 1/5th of the hashpower, I as a user have to wait longer to get my tx into the chain.
Now something similar is already happening, and has been for a long time. I can pay a pool to get my transaction a higher priority. Instead of replacing my transaction with a higher fee, i pay the miner directly so they will include it in their next block.
I get it you are thinking allong the lines of ''well if this pool has a part of the mempool that is only exlusive to themselves, they will get that money, but others wont'', yet from a user perspective, it makes no sense to do this. It takes more time, i see no reason why it would be cheaper.
Remember, the systems functions under ''I as a user am indifferent as to who includes my TX into a block, as long as it happens'' & ''I as a miner am indifferent as to what TX's I am including, as long as it has sufficient fee's''.