I did unscrew everything. Though, still sharing your ideas for fixing this in case anybody else is looking for more ideas is ok, of course. Below is a quote of when I stated I fixed it in case anybody missed it.
Keeping the drill bit centered on the back of the standoff was a challenge. I successfully drilled out all of the standoffs but damaged the threads in two of the holes of the motherboard tray enough that the replacement standoffs are loose and I need a couple small nuts to secure them to the tray. After I separated the motherboard from the tray I used vice grips and a screw driver to unscrew the screws from the brass standoffs.
That worked for me but I didn't have a wide variety of tools to work with. The other suggestions may work very well for somebody else.
I mentioned I damaged the threads in two of the holes in the motherboard tray. Two brass standoffs are a bit loose in the tray. I was looking for some small nuts and maybe lock washers to put on the brass standoffs to help secure them to the tray but these standoffs have fine threads. Looks like a 4-40 sized nut should fit but the thread on the standoff is more like a 42 or 45, maybe. Somebody at Home Depot, Ace Hardware, and Radio Shack helped me look for a nut that would fit but we couldn't find one.
This is an old motherboard and case so chances are low that the case will be needed for any other motherboard. Maybe I should just use epoxy and glue the two brass standoffs to the tray.

Or, if I could find an old broken optical disc drive that has matching threads I could cut out the threaded holes from the metal shell and make the cut out parts in to something like a nut. Recently I just tossed out an optical disc drive that had a burned out circuit board but I didn't think of this earlier and now I can't get the drive back.