I also do not understand why "you need to share the source when you expose" is apparently so much worse than "you need to share the source when you distribute".
I also do not understand why "you need to share the source when you expose" is apparently so much worse than "you need to share the source when you distribute".
Because distribution is an act controlled by copyright, while usage (which includes exposing) is not. That Free Software licenses are pure copyright licenses is a fundamental tenet of Free Software (which follows from Freedom Zero). We call licenses that restrict usage EULAs. You know, like the Windows license.
Thankfully, the AGPL isn't actually an EULA. In turn, it doesn't effectively accomplish its goal of avoiding SaaSification. Instead, it tries to work around this fundamental issue by imposing restrictions on how you can modify the code instead, therefore making it not Free Software, since such restrictions violate Freedom 1 of the Free Software Definition. Kind of like some of those weird nonfree "shared source" licenses.