While reading updates on Parallels Desktop, I saw a mention of another desktop package called VirtueDesktops. I sucked it down and installed it, and lo and behold, it almost fails to suck.
VirtueDesktops is built on top of Desktop Manager, and fixes all of the bugs that had been irritating me. In particular, it doesn't leave a click hanging in the menu bar after switching screens, it doesn't do a screen swap animation when no screen switch is actually occurring, and it shows the notification bezel on the screen being switched to after the animation.
There's an even bigger advantage, though. There is no pager on the screen at all times (though one can be brought up by keystroke). It's not needed. Switching to another application in the Dock will swap screens to that application, if needed. To me, this is the Right Answer.
It's in active development, and some features aren't implemented yet, but it's a winner even now. Recommended.