The first time a new transaction sent is not important. Such payment can be lost, disappear or part of doublespend scam.

Bitcoin can operate correctly when only block travel on the network and everybody will not see a difference.
Users can send payment to his miner by any other method such as email, sms, website. It's not really a problem.

The serious business is actually sending blocks, not payments. For miner it is very important that everybody in the world can see the whole block including all payments because that allows other miners to immediately check if every rule is correct. The other miners must do this check because if there is problem like payment with bad signature(scam), they all lose reward funds (not just the first miner) and mining power got wasted.