Bless you - this process worked like a charm. (switch Tiny MCE to "Extended" in "Plug-In Manager; then in Tiny MCE click "Insert Media" and plug in the URL to the You-Tube video, not the embed code.)
A couple of additional observations may be helpful:
1) It took me a while to find "Tiny MCE" in the Plug-In Manager - finally found it there, listed as "Editor-Tiny MCE".
2) Also, took a while to find the "extended mode" referred to. Finally found it at the top of "Plugin Parameters/Functionality"
(the default is "Advanced"; just toggle it to "Extended.")
3) Several experiments seemed to indicate that the "Insert Media" process ONLY WORKS FROM THE BACKEND.
From the front end, the "insert media" tab appears, but the video isn't saved. I found this out when I successfully installed my video to an article on the backend, as described. Then I tried to edit some text in the article on the front end, saving the edit - and the You-Tube image disappeared! Once I installed the video again from the backend, it reappeared. This seemed consistent after several tries.
4) "Extended" about triples the functionality of Tiny MCE!
I was going to download JCE for expanded editing functions - maybe now I won't have to. So, thanks for the tip - both for the You Tube question, and for the "expanded" editor functionality. Had no clue -this is SO much better!