Tree root problems
This question has a couple of potential complicating factors, but since the trees were yours and not a neighbor's it simplifies them.
Since the trees were on your property and you no longer want them you can use brush control herbicides to control them. If you are in the county you may be able to get chemical at a reduced rate from the Weed & Pest District through their cost share program. But you wouldn't want to use this method to control suckers from a neighbor's living tree or from a tree that you have and still want to keep. The chemical would likely translocate through the roots to the main trunk and kill a living tree.
You need to be sure the species you want to control is listed as a target species on the label and of course use the product following label directions.
If you want to go with a bit milder solution you could purchase a product that controls suckers that come up from roots or the base of the trunk of trees. It is called sucker stopper or sucker eliminator. This is a plant hormone that stops or slows the suckering process but is not a permanent solution to a frustrating situation.
It looks like you have been digging the suckers up at this point and using the sucker stopper or sucker eliminator product on the cut roots will help to prevent new growth at the damaged root sites.
Eventually nature will take over and you will probably get some fungal growth on all the decaying root material and may have to deal with fairy ring in the lawn unless you decide to excavate the roots now and eliminate the food source for the forthcoming fungal colonies.
If you need more information please feel free to contact the local office at (307) 235-9400.