In my personal opinion, AGPL is largely chosen to avoid various cloud providers from profiting significantly from $product, without ever giving back. That's control, but a very specific form of it. I don't personally like AGPLs legalese, it's very very imprecise.
Maybe it just seems imprecise to a layperson because they don't know the well-defined meaning of various legal terms? (... and confuse these with their fuzzy meaning in ordinary language)
You're sure they were talking AGPLv3 and not [L]GPLv3?
The definitions of what constitutes an interactive program is quite vague (sect 0 and 13). Let's say you have a database server under AGPL (mongo, or say citus). Clearly they support interactive access in some form, but from the perspective of user of an application using said database access is not interactive, nor is it clear how the database could provide such an interactive notice. Various vendors addressed that issue with clarifying notes about their understanding, but that definitely increases doubts of possible users, including their lawyers.