Oh, jeez.
In the properties, set "Fill Color" to "Background". No wonder I never saw it. I was busily looking for a property that was easily-understood, like "Background mask".
Gotta put on my Autodesk cap, and think about things in a "skew" fashion, and then it makes sense...
Use either DIMTFILL or "Fill Color" to turn on background masks for dimensions. I would have found this years ago if it was clearly stated in the Help, or if it was named something better, but I got it now. Better late than never, eh?
OK, so now how about getting to the text component of MLEADERS via .NET? Anyone tried that? Had any luck?