Upgrade and migrate SQL Server 2014 AlwaysOn AGs to SQL Server 2016, using the existing WSFC name
Ah-ha!! I figured out what I did wrong. In my first attempts, I was doing all the work from the existing primary (SQL2014OnSite), and when I removed it from the AG, I must have done it in such a way that it blew away the AG. This time, after failing over to the new primary node (SQL2016OnSite), I made the change on this instance by right-clicking the AG, choosing Properties, and in that window, highlighted SQL2014OnSite, and clicked the Remove button. It has now been running in the new configuration, which is SQL2016OnSite as primary and SQL2016OffSite as secondary, for about 15 minutes after removing SQL2014OnSite from the AG, and...so far, so good. I think that was my error: performing the "after-migration" work from the 2014 instance instead of the 2016 instance.
Still, if anyone has any insights or recommendations for a better way to do this, by all means, please suggest away.