If you buy Bitcoin through P2P, you may use cash, gift cards, or prepaid cards. You may also purchase via over-the-counter with cash. There are also Bitcoin ATMs that only ask for your phone number when buying. You may use a burner number for that. You can also opt for meet-ups in doing transactions. Again, with cold cash, of course.
All of these options would keep your identity private. However, you may appear personally or be recorded in a camera. That doesn't necessarily mean you reveal your identity, though. You can always do something to hide your real face. You can buy a 1$ mustache, for example.
Yes exactly we can use cash or gift cards or prepaid cards to buy Bitcoin without linking it to bank account. Even Bitcoin ATMs let us use temporary phone number for privacy. However even if our financial details are hidden but person seen on camera means someone could potentially figure out who we are as your funny idea of $1 mustache shows. So while we can limit what is known about us online being physically present always adds another layer of things to think about for privacy.