What if the computer you want to log in on doesn't have Bluetooth? Probably most public computers (like ones in libraries) don't have it.
Only if they’re a bit old. Nowadays WiFi chips double as Bluetooth chips on newer platforms.
My suggestion would be to not do that. But keep a password and an offline TOTP app if you must. It’s still an option.
The website should display a qr code you can scan with your phone that allows you to then login, unfortunately a lot of sites don't implement this, and some don't implement backup codes. This isn't the fault of passkeys per se, but of poor implementations.