In my opinion, no website should be used to send transactions at all. The best way to send a secure transaction is through the Electrum wallet, so the transaction won't be stolen and there's no need to worry about the delivery.
Electrum will use public pool to submit your transaction. And because it's public, everyone can see before it's confirmed and so everyone can steal if the private key uses low entropy like in this case.
And no, disabling RBF won't do anything at all.
This is why it's suggested to use private pool to submit your TX. It's not visible until mined/confirmed.