You can transfer money from your checking account to Cash App, and then use your Cash App balance to buy bitcoin. That's what I do. I'm not really concerned with total anonymity for my bitcoin purchases. My purchases are always under the reporting threshold, and I immediately move my coins to a non-custodial wallet. I won't keep my coins in the built-in exchange wallet.

Someone who wanted complete anonymity would be better off using Robosats. You can buy with Amazon eGift Cards, Tether, Zelle, or Wise. Even though some of them are KYC methods, they're just cash transfers and aren't linked with whatever you're buying.