I can finally chime in.
Last weekend, hubs took our 2400-pound mini-excavator with an 8" toothed bucket to see what he could do with the little mesquites resurfacing on our 10 acres in Burnet, TX. I was pretty sure this place had been bush-hogged before, and I'm more sure now. These "little" mesquites 2-4 feet tall up to 3/4" in diameter shoots/trunks all had a big knot a few inches below the soil's surface (fist-sized) and a long root 1-2" thick that stretched horizontally - very interesting finding that. So these were not saplings at all. Luckily, we have only a couple hundred of them I think, and they seem to grow back from roots slowly. Hubs had us cut the knot off and cut that horizontal root and he thinks that will do the trick. We'll see.
We have many small trees that have just a few trunks so we'll keep those to turn into decent mature trees - maybe 5-10 of those.
There are maybe 30 medium sized super bushy, wispy 5-8-foot tall mesquites. which means in the past they've been knocked to the ground - they have 20-30 trunks - am not quite sure what we'll do with those but here's an option that is interesting :
https://www.youtube.com/watch?v=a2uQpPgFj9U - November burn experiment, near Fredericksburg, TX
Spring update - none regrew by spring:
https://www.youtube.com/watch?v=9F2GNLm1Ix8
With a HUGE number of mesquites to get rid of, I think I would bush hog them just to get a handle on them, then either over time burn the stumps, or dig deeper and cut the roots.