OK. I eventually tracked it down.
Before adding the intro element, I started with the menu. When I added the intro movie, all looked good on the flowchart and it played OK in Encore, but, as I explained initially, when burned to DVD playback started with the old menu, not the new first play.
I tried deleting Encores preferences (start with SHIFT+CONTROL) to no effect.
I tried adding STOP at the end of menus, to no effect.
I also tried a variety of other first play items, but the DVD always started with the non-first-play menu.
Eventually I looked at the project properties and found that the title button and main timeline were still pointing at the original first-play menu item, not the one currently selected. Changing the main timeline had no effect, but changing the Title Button to the new first-play timeline finally solved my problem.
It seems possible that there is an issue with updating all relevant items in Encore CS6 (Windows, fully updated) when changing first play. Or am I missing something that makes it my fault here?