Accelerate decomposition of small tree stump

Buy one of these

I used a mattock for 20-30 of this sized "volunteer" tree roots. For the larger trees (6" diameter and up, a couple approaching 20"), I used either a bow saw or a chain saw to cut, then used a mattock on some after digging around the stump to reveal roots radiating out. For the extremely large stumps (including the 35"-40" stump left by the city after they cut down trees interfering with power lines), I had a guy with a grinder come in. He used a large unit, not a bumber mount, which was self-propelled.

If they aren't too big, digging/cutting/roots/digging/cutting roots around the tree works pretty well to get the stump below ground level at least a foot or so. Once you get the first couple roots cut with the mattock, additional roots become easier to cut.

Or weed and root killer.