Sending transactions to mining nodes only? Instead of propagating transactions across all nodes in the bitcoin network, would it be theoretically better to just send transactions to mining nodes as transactions are only confirmed and 'used' when confirmed inside a block. The other nodes would then be able to confirm and validate these transactions when they learn about the new mined block with the 'confirmed' transactions?
In theory, that would be ideal if your goal is solely getting the transaction to mine.

Hurts permissionless mining However it does require knowing how to reach miners directly. Mining is designed to be a permissionless business, accessible for anyone with the right hardware. Your solution would require publishing the IP addresses of miners to all transaction creators.

Hurts sender privacy It would also hurt privacy, as miners would learn the sender IP of transaction creators; possibly enabling them to censor based on that information.

Not relaying transactions doesn't gain you much The marginal cost of relaying every transaction to every node is low since the introduction of algorithms like BIP152 (compact blocks). Assuming a transaction will eventually be mined anyway, that transaction needs to reach every full node eventually (as part of a block). Relaying it as a separate transaction ahead of time doesn't cost anything, if the relay of the block can refer to the earlier transaction instead. There are some costs related to 'inv' messages that can be reduced though, without removing transaction relay.

Hurts block propagation speed And lastly, only broadcasting transaction to miners directly would worsen block propagation speed very significantly. Modern block relay protocols (like BIP152 and FIBRE) rely on nodes knowing the bulk of blocks' transaction ahead of time to quickly relay those blocks.