The third party doesn't pay back the broker, they flat out pay them a fee to send transactions through their systems. The broker doesn't allow or deny any transactions, just in the absence of contradictory routing instructions they send their default orders through the third party.

The customer is usually none the wiser as they are being charged the same fee from their broker typically regardless of the routing. The only way a customer would know is if the broker discloses it to them. In the US equities markets that disclosure is a legal requirement.