From what you describe, I'm thinking that it's probably a Quince bush. (Have you tasted the fruit? It should be
extremely bitter!) I have one at the edge of my yard that was so out of hand when we moved here, that I cut it way back (One branch at a time!) ...& it came back ...with a vengeance!
That darned thing will get 20 feet tall in one year, & as you say,
"It sends off shoots that go underground and they spring up all around the diameter of the shrub- it covers the ground under the shrub. Even these offshoots have small thorns. It's proved to be a very hearty shrub- sort of the shrub that won't die." Last year my son took a chain saw to it & cut it all the way to the ground... & it's coming back again!
I don't like using chemical herbicides, so I plan to just keep cutting it down & hope that eventually I'll weaken it to the point where it'll give up!
Several years ago we had a Quince Hedge that was real nice, but we had to keep it shaped & trimmed, and those offshoots that come up from the root system had to be chopped out constantly.
I'm sure that since those offshoots can be controlled by chopping them out, you might also be able to eliminate the bush itself, by chopping/digging into the root system. The only thing wrong with that, is that some
plants will send up new growth from each piece of root that's left behind! If you're in an area where you're allowed open burning, maybe a bonfire in that spot, after you cut the roots out, would do the trick.
I'm sorry that I don't have a better answer to your question, but I can certainly relate to your frustration over this nemisis shrub! As I said, I don't like using herbicides, but sometimes there is just no other answer! I suggest that you call your County extension office & ask them what they recommend. They usually have Master Gardeners there, who are qualified to advise people with this type of problem.
Good luck! (I know I'm not the only one here who would appreciate hearing back from you when/if you find a remedy for getting rid of that horrible bush!)