I tested 5.20 from Shuduo’s PPA and for me it reproduces the issue just the same as -proposed. To be sure, I then went back down to release versions in all the other packages (so that the only change wrt. released is again just Unity being 5.20, just now from the PPA), and the issue remained. Downgrade Unity to 5.18 and it again goes away.
But this looks like a speed issue, so Unity may not be directly at fault: any apps started once the desktop is fully loaded open as truly maximized, so a workaround is setting autostarted apps to start after a delay. Perhaps Unity going from 5.18 to 5.20 here causes just enough delay for those apps to start before whatever the component is that should fuse the menus with appindicator. The autostarting facility should wait for that component, or the component should be also able to fuse menus for existing windows once it gets up and running.
Incidentally, any appindicator-related packages’ versions also don’t seem to matter for this once Unity is 5.20. I think I’ll try upgrading just those to their -proposed versions next (so that Unity remains at 5.18) just to cover all angles.