Then, sorry for referring back to the same example again and again, how does Blockchain.info broadcast it without it's peers telling it "no"?
They are extremely well connected, probably thousands of connections to the network, whereas you only have a dozen or so. So it is likely that a miner that does not care about the fee is listening to what blockchain.info is transmitting, even if the majority of the network does not propagate it.
(I am speculating here)