The AGPL does not just say he has to have access to the modified software. It also says he has to be granted permission to redistribute it, or derived works, under the same terms.
He redistributed a derived work under the same terms and got hit with the threat of legal action.
I don't know what "access my garage and use all my tools" is supposed to be an analogy for in this situation.
He is free to remove the part where it spoofs itself as BambuStudio and do whatever the hell he wants with it. He can probably distribute it with the spoof and continue without actually getting sued because they are likely fixing this as I type this. If they do sue him it will be for unauthorized use of their cloud services. I do think they'd have to sue individual users.
Accessing my garage and using my tools == using Bambu's cloud infrastructure, which they clearly do not want him (or apparently any non-BambuStudio clients) doing any more.