The reality is still that "it's your responsibility" does not exclude the concept of "so you have delegated that responsibility to your other admins."
I think the key interlock here is that the purpose for phpBB removing the database backup download option was to ensure that "something" other than just phpBB itself would have to be compromised in order to gain access to the entire phpBB database. Since that exposes things that even a founder account cannot see through phpBB's own interfaces.
So "delegating database backup" in the face of this phpBB intention that "something besides phpBB itself would have to be compromised in order to see the backup" pretty definitively means that you must create something similar to the "FTP account with read-only access to the /store directory" like warmweer described.
Such that no, we're not talking about you handing over the keys to your hosting account access to anyone you didn't already trust with that information. But yes, there is now "something besides phpBB itself" (the new additional FTP account credentials) that would have to be compromised in addition to phpBB itself in order to trigger a current backup & skulk away with the entire current full database.
Certainly you could approach it differently, or through an even more elaborate design. But that seems like the minimum of what's in line with the intention, and for "what must I do, in response to what phpBB has done."