Miners should send a little % of the fees of the block to all connected peers. Independently of the protocol.
Why would they choose to do that?
This has some MIM attack problems isn'it?
Yes, but even if that was resolved --- All that would seemingly do is incentivize people to sybil attack the network.
As a incentive to run a full node.
Full nodes secure the network too, why is there no incentive on the protocol level like the mining process?