my concern with this class of thing is that it just becomes a way to abuse the protocol by a few actors at the expense of all users of it.
for example, I make a contract and construct it to pay you out of the burned fees for interacting with it. I add token mechanics on top which I use to vampire attack a popular DeFi protocol to juice the yield even more.
I use the remainder out of the aggregate flows to bribe proposers to prioritize transactions to this contract ahead of all others. I even give miners a bit more in the form of kickbacks to start raising the gas limit to include this “parasitic” transaction over all others.
Assuming this construction doesn’t collapse, you’d end up with pressure to raise the gas limit meaning fewer people can run full nodes harming network security.
If you follow this line of thinking to its limit, you see why we burn the base fee in the first place – it is the least capturable way to handle what are otherwise valuable flows.