Best way to remove roots around a tree stump

A woody plant killer like crossbow is the only way I know of and it won’t hurt the lawn. I usually paint the stump just after cutting a tree around the outside just inside the bark with concentrated crossbow and let that sit for at least a few weeks. If any shoots come up hit them with crossbow it won’t kill the lawn unless it is all dandelion’s. This has worked for me on even cotton wood that will continue to put shoots up for years if not treated when cut down.

You have a couple of options.

Dig out the roots by hand which is an arduous task. There are specific tools for digging out roots to make it easier to leverage out the residual roots.

The easiest option is to remove the surface layer of soil and expose the remaining stump or roots and apply either a chemical root rot or natural substance like Epsom salts. However, there is a specific process to follow for these treatments to be effective take a look at this guide Remove Tree Stump Roots (it has a video too which is useful)

Once you have completed this, it is best to cover with a layer of weed or light suppressing fabric to make sure you don't get any further shoots.