Thanks to the tip from squane about repopulating the library, GENIUS IS WORKING AGAIN!
It does seem to be related to having started iTunes Match. My guess would be the songs that did not make it to the cloud may be the culprit, but I'm not sure.
The steps are a little involved, but here's what I did:
1) Go through the iTunes Match process
2) Click the cloud icon in the sort bar at the top of itunes
3) Select all the songs that were matched (if you're like me, you'll have about 50 songs/booklets that did not make it to the cloud for various reason -- so it's important not to select these at this point).
4) Delete, and, of course, do not select "also delete this song from the cloud" (I'm doing this because I have a MacBook Air with an SSD drive, so I can't afford the space to have my 94 GB of songs on my drive).
If you haven't tried this before (only keeping the cloud version), you might try it with one song just to test. After the local version is gone, you still see the song but you need to be connected to the net to listen to it.
Now, all your songs, except the ones that did not make it to the cloud, are off your local drive.
5) For all the things you want to keep, but did not make it to the cloud, I right clicked on each item, selected "Show in Finder", and dragged it to a folder on my desktop so that I'm sure not to lose it in subsequent steps.
6) Quit itunes
7) Go to your Music/iTunes folder
8) Delete all files that begin with iTunes (so iTunes Library.itl, iTunes Music Library.xml, * Extras.itdb, * Genius.itdb)
I'd recommend backing up these files somewhere before deletion just in case...
9) Restart iTunes
10) Turn on Match
11) Turn on/update Genius
At least for me, this process fixed Genius being unavailable for any song I tried, even those songs that were the basis of previous Genius song lists that I have saved. It still shows as unavailable for certain songs like before, but all the songs it previously worked for work again.