If you sync your secrets to some cloud service then yes, you are trusting that cloud service. And if you let your TOTP cloud service reset your account with an email then it probably isn't the most secure option.
But the important thing here is that the user is in control. They can memorize their secret if they want to an no one can take it from them. Or they can publish it online if they don't like security. With SMS 2FA you need to trust your telecom provider, I very much don't.
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.
There was a DND that talked about how sim swaps used to be a cakewalk over the phone social engineering exercise but were now kind of expensive to pull off and required a man in a physical location, with T-mobile remaining the easiest target. The black hat guy they were talking to said his first steps were finding a target worth swapping, usually some one that bragged a lot about bitcoin or some other crypto currency on twitter. And getting the phone number was usually really easy to do with a combination of OSINT and abusing the fact services will give you a partially masked phone number when you try to login.