OK, just in case someone else comes here seeking an answer ... It seems they finally (at least for whatever issues I was having) fixed the problem.
Here's the last email I got from support (end of message). I followed the instructions and now seem to have a working version of the Creative Cloud app. After all that, though, I'm not sure it was worth all the heartache. Downloading the applications one by one from the prodesigntools site seems to make me feel better for some reason. Anyway ... haven't messed around with the CC app yet too much, so I'm not totally convinced it's "all" fixed. But just the fact that they actually seem to have fixed something makes me feel a bit better.
From adobe:
Just wanted to follow up on the same conversation that we have been having around downloading the Creative Cloud Desktop Application . I would like to check if we ever were able to download the product after the last discussion that we had or not . If not , Can you please try and use the following steps and do let me know the outcome please.
1) Navigate to adobe.com/downloads .
2) Go to Other downloads and select the download for Adobe Application manager for Windows .
3) Once the download gets completed , please launch Adobe application manager and that should update itself to Creative Cloud .
Kindly let us know if this works or not for you.