Aha. My problem is something to do with my configuration settings. When I go with the default setting “Still_Draft”, everything works as expected.
The “My Favorite Settings” config I made based off the tutorial is, at least checking cursorily, correct, though I will check against the ArchViz project to see for sure.
I suspect the error might be that my configuration settings should include a Game Override setting, like Still Draft does, which might explain why it seems like the Game Mode might somehow be interfering with Sequencer.