Kraken is another US facing exchange (full KYC compliance with selfies and ID and shit required) but it tends to shit the bed at times of high price volatility such as the big DOGE pump around the time of the Gamestop thing, meanty I couldn't take advantage of that.

Don't know of any anonymous ways to turn cash into BTC anymore, not like the good old days

But reiterate what Dan said, it doesn't necessarily have to be blockchain.info, any non exchange wallet will do. They key is you can't go directly from an exchange (a site that lets you trade between fiat and crypto) to a gambling site. It needs to be through a NON CUSTODIAL WALLET (one that you hold the access keys to, not the site).

Funds in a non custodial wallet cannot be taken from you unless you lose your private key or someone discovers it. Funds in a custodial wallet can be frozen if someone decides they don't like your account for whatever reason (gambling, political donations to the wrong people, etc.)