Well, restoring from iTunes didnt work... at all. Despite trying all of Apple's prescribed troubleshooting steps, iTunes would show the AppleTV in the sidebar, but would continually insist that it couldn't be found. Luckily, for me, I'm on the phone with Apple technical support, and it appears that the (very nice) support rep, that is helping me, has seen at least a few instances of both my problem, in the last few days. Multiply 'a few' by the number of support reps that Apple has, and this qualifies as a bigger issue. According to the rep, one or both of these issues are due to the iTunes 11 and/or AppleTV 5.1.1 update. We're trying to back down iTunes to version 10.7, so that we can then back the AppleTV down to 5.0.2.
Note for it you have to go through this process: Once iTunes 11 has been uninstalled, and iTunes 10.7 has been re-installed, it won't want to launch, because the iTunes library was made with a newer version of iTunes. To get around this, hold down the alt/option key when launching iTunes. This will present you with the options to create a new library or choose an existing library. Choosing the existing library will get you the same error about the library being made with a newer version of iTunes. Choose the option to create a new library, and let that go. Make sure you name your new library folder something other than just 'iTunes', or you may overwrite your existing library. You may see a couple of odd error messages, but I believe that, in this case, they are incosequential. Once iTunes 10.7 has launched, just use File -> Add to library, and choose the old iTunes folder (in your Music folder).
(That should all all of your existing music, movies, etc., to the new library... I think. I'm not a Mac (or software, in general) neophyte, but I'm not an official Apple support technician, either, and I don't have access to the same database of info that they do. Luckily, importing all of the old files into the new library is a non-destructive process - meaning that the old files stay where they are and should not be modified in any way.)
I got iTunes 10.7 reinstalled. I'm getting a couple of error messages when I launch it, but it does launch. Also, it now acknowledges the existence of the AppleTV. 'm downloading the older version of iOS for the AppleTV, and when it's done, I'll do the restore. If/when I can confirm that's working correctly, I'll likely wait until a new software patch is released for the AppleTV, before updating it again. I think that AppleTV OS 5.1.1 and iTunes 11 are meant to work together, so I think I'll wait on updating iTunes, again, too.