If you need to buy Bitcoin anonymously, a Bitcoin ATM should not even be an option for you, because you are not buying it privately; your primary data will be collected by the ATM, which is against what you are looking for.

The best way to buy bitcoin and remain anonymous is to use an exchange, but not the popular centralized exchanges that you are aware of, but P2P exchanges where you can discuss with your seller which payment method you prefer the most: cash, gift cards, etc.