El Capitan Mail 9.0 won't accept active accounts
O.K. I have an update. I had checked the "hand-off" to another Mac in the General preferences. I unchecked it and now have two of the accounts added. Then, finally, I unchecked all the Google items in the Internet Accounts pane, turned off any device with Google anything, logged into Google from the Internet Accounts pane and mirabile dictu! gmail is now in place and downloading messages.

So: 1. Handoff is a nice feature, but it sure can mess up Mail.
    2. Gmail is as quirky as many have claimed.

Thanks to bike for his answer.