What about payment ids - they are not used at all. Once your deposit XMR address is generated, it is used without payment id. It seems a bit danger to me, because as I know I has no "official" ways to prove I made a transaction to that address.
Are they providing an integrated address (106 characters) for deposits? Because those still allow one to verify that his transaction correctly arrived. Basically you can use this guide with either a plain address, a plain address + a long (old), unencrypted payment ID, plain address + short (new), encrypted payment ID, an integrated address, or a subaddress.
https://monero.stackexchange.com/questions/6134/how-do-i-as-a-sender-verify-that-my-transaction-actually-arrived