This morning I tried making a new mail account of type IMAP from Mail's Preferences panel.
Now, in order to avoid the Mail program's "help" in doing this, I chose "Other" as the type and put in false account information during the set-up step. This was so that Mail wouldn't know that this was actually a GMail account.
Then, after it was set up (and of course, didn't work), I changed the settings on the account to be the right ones. I changed the incoming mail server and the passwords and all the "Mailbox Behaviors" to be GMail's suggested settings.
The account appeared and shows up as "IMAP", not "Google IMAP".
I am now testing, but I am hopeful that this will at least put us back to a state where it's a regular IMAP account and therefore no extra funny business. At least we can predict what will work and what won't. Maybe it will act just like it did pre-Mavericks.
One snag, I found that even though I had unchecked "Move deleted messages to the Trash mailbox", that option got re-checked when Mail finally connected to the account. [I have noticed in the past these options mysteriously getting re-checked, but I have never figured out why or when that occurs.]
Anyway, that's what I did this morning. I'll continue testing.
Bark!