In comp.periphs.printers, wrote:
They are called "Booklet Makers" or Folder Staplers and they used to
start (20 years ago) at $4000. That's adjustable from tiny to 11"
wide, from tiny to 17" long, and up to 24 sheets. Try goggling
printers parts if you can't find it directly. Smaller models, if
they exist, for fixed bookletting 8.5 by11 only, with fewer sheets are
probably available as they are used by schools and non-profits.

Searching with that phrase ("booklet makers") instead of the "saddle
stitch" phrase I had been using does give different results. I see
for example the Bookletmate:

http://www.generalgraphic.com/bookletmate.htm

At $700, the price is within my budget, but the feature list isn't so
long. It staples and folds. It's not doing the collating (which for me
works out to separating every N pages and adding the cover sheets, which
are printed separately on different paper). So that's not quite enough
to be tempting.

But thanks for pointing that out. It might become tempting if the
workload continues to increase.

Elijah
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but also just getting a service to make them will be tempting, too