To be honest: A cautious player won't trust you more because you provide them with the private keys to their deposit address, especially knowing that you store a copy of the private key on your own server anyway. If anything it opens up additional attack vectors for hackers and puts the balances of your players at risk. To make matters worse, even without a hacker attack any player could move their own coins and then throw scam accusations at you for not keeping their coins save. There's no way to solve such a dispute (ie. whether you stole the coins or whether the player moved the coins themselves).


Short of falling back on some of the less widely used alts I'm afraid you'll only have two choices:

a) Accept payments directly on a cold storage address and keep track of in-game moneyflow on an internal database, off-chain. This will require the trust and goodwill of your player, but as mentioned above: Even providing the private key to their deposit address won't replace trust.

b) Become a bleeding edge pioneer and start accepting LN micro transactions. (YOLO!)
https://medium.com/@ismailakkila/bitcoin-setup-your-own-lightning-node-on-mainnet-94337bda09fa