I meant “download the totp generator”. Correct me if I’m wrong but the totp generator has to linked with some account, the security of which is not managed by you, and is just as vulnerable as that account
You may be confusing TOTP with proprietary app-based 2FA solutions which just send the token in a push notification or similar.
The system is designed to assume that the user doing the initializing that one time is the legitimate user.
(There's a bootstrapping problem for the authenticators, that the account provider needs to "trust once" that the user is legitimate. The best time to do that is as early as possible. Preferably when the account is set up, and before much value/dependence has been invested in the account.)
After that first initialization of an authenticator for an account, anyone trying to initialize another authenticator would have the burden of trying to prove to the account provider that they aren't just an illegitimate person trying to bypass the 2FA.
So, you probably can't just use email to do a "lost my authenticator lol", unless the account provider doesn't really care about 2FA, and has implemented it in a very weak way.
As such, no, it's not associated with anything from the source account. It is not challenge/response, and you can scan the QR code with 10 different phones and they will all produce the same codes at the same time.