What makes no sense is sending a payment doesn't require you have the blockchain fully downloaded and updated. You just have to broadcast the transaction.
Most BTC companies require the ability to generate new random addresses on the fly for deposits, and what not, as well as a wide range of other functions that are accomplished using API commands directed at full node.
That's the rub though. We aren't talking about deposits, we're talking about sending the bitcoin from the BT account to the people who placed withdrawals. Sending from their wallet, not receiving on their wallet.