Ask Mr. Smarty Plants

First and foremost, NO ROTOTILLER!! That would almost surely do some damage to the tree itself, and to the roots of the Quercus fusiformis (Escarpment live oak). This is not the time to be doing any kind of damage to a tree, and most especially to a Live Oak. That damage would then make your tree very susceptible to the Nitulidid beetle, whose line of work is to visit the sap from damaged trees for a bite to eat. He doesn't really mean to, but he is a prime spreader of Oak Wilt, the bane of all oaks, but most especially of Live Oaks and Red Oaks, and even more especially in the Austin area, where arborists all over town are struggling to contain it, as there is no cure. When the beetle discovers the lovely sap you have provided him, he will drop in for another bite and, since he is carrying the fungus on his body that causes Oak Wilt from the last eating establishment he visited, he will infect your tree. This beetle is active in all but the very hottest and very coldest times of the year, which is why we recommend pruning only between about November 15 and January 15.

You are not the first person, nor will you be the last, to ask us about this problem. It is a trait of the Live Oak. In the wild, the tree will eventually form a motte, which is all those little sprouts (well not all, they won't ALL survive) grown up to be Live Oaks, too. This is not good, of course, in a residential situation, and, worse, Oak Wilt can travel through those intertwined roots-one tree in that motte gets infected and they are all infected. If the oak tree is in grass the sprouts can be mowed. This won't kill them, but it makes it look nicer. .

From a previous Mr. Smarty Plants Question:

Sorry, but the only safe way to get rid of those sprouts is to cut them off. You can dig down and cut them off below ground. They will still return but it will take them a little longer to come back out again than if you cut them off at or above the surface. Any sort of herbicide you might use will affect the growth of the parent tree and I'm sure you don't want that to happen. According to John Begnaud, horticulturist, writing in Go San Angelo:

"To date, there are no repellants, hormones or chemical sprays that reliably suppress or remove thse suckers without harming the mother trees. Hand grubbing or deeproot pruning can reduce these suckers for a few years at best and then they will return."

You can read the answer to a previous question concerning the problem of live oak sprouts.

Now, as to your disagreement with your landscaper. Who's in charge there? If he can't, or won't, clip off those suckers, get yourself a long-handled pruning nipper, so you don't have to crawl along the ground. Get as far below the soil as you can, and clip off those sprouts. Just keep after it, make it a habit, because they will come back.