We don't currently propagate weak blocks but there have been proposals to do exactly this to solve the compact blocks issue I linked to. Additional reading: https://delvingbitcoin.org/t/second-look-at-weak-blocks/805/1

Also, decentralized mining pools like P2Pool and Braidpool track weak blocks in a blockchain (or block DAG in braidpool's case) in order to reach decentralized consensus on share accounting.

We need decentralized pools anyway, why not also use their weak blocks for fee estimation and compact block propagation? It's not a perfect signal but more data usually leads to better estimates.