OK, I think I finally stumbled across the Open Sesame command, through pure dumb luck. I finally tried Tabs-Close Tab. When I closed and re-opened WinSCP, the remote site was automatically connected and displayed in the right side panel.
Maybe it’s just me, but I find the WinSCP interface incredibly confusing. At the very least, this technique needs to be much better documented. Perhaps a “Common Problems” Web page is in order, explaining how to resolve this issue.
I still don’t understand why this problem occurred when I merely upgraded to a new version. The show Login dialog on startup and when the last session is closed was always/still checked, so that wasn't the cause of of the problem. That makes me worry this is going to recur every time I upgrade.