I wrote to the FSF's licensing team about this question:
[...] Does this [section 13] mean that if I run a *completely unmodified* AGPL-licensed program as a network service, I am *not* required to offer the source code to network users?
And I received this response (bracketed phrase added by me):
[...] If you haven't modified the software then you are not required to add that functionality [i.e., to download the source]. Of course, if the functionality to download the source is already in the unmodified software, it will already be there for everyone to enjoy.
So, if you use an unmodified AGPL application that doesn't have download-source functionality, you are not required to add one or otherwise offer the source to users. If you do modify the software, of course, you are required to add a mechanism to allow users to download your modified source.
As a practical matter, an author who cares about source-sharing enough to license code under the AGPL would probably include a mechanism or link to download the source in the original program. This is kind of an edge case, because it only applies when both (1) you want to use the AGPL software unmodified, and (2) the AGPL software doesn't already include a download-source mechanism. If either of those conditions is false, the software must (or already does) include a way to download the source.