If it were just "running on your server", there would be no distribution
requirement. But it is running on your server and sending and receiving
data from the user, which is different.
This is the core of the issue. If you are a local user of the software, the AGPLv3 is identical to the GPLv3. It's only when you're running the software on your machine for other people to use (whether that be an IRC bot, a webapp, or a game server) does the AGPLv3 specific clauses take effect.
In these cases, all it's doing is ensuring that the users of the software are granted the four software freedoms. We do not view this as a use restriction, as the user of the software has no restrictions added to her remote usage nor local usage should they download it, but rather ensuring that she has the same abilities and rights we have with locally-run free software.
If it's a small embedded system, the source code is likely also to be small.
An excellent point.
And if you can't afford the costs of the bandwidth for the small
embedded system, you can't run the service at all! Free as in freedom
does not necessarily mean free as in cost to you.
.. and even if hosting the source code over your own Internet connection, it should also be noted that in almost any case remote users downloading the modified source should represent an extremely small part of your bandwidth.