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?

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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?