Sunset Tree uses a machine that literally chews up stumps 6 to 12 inches below ground level. A set of carbide teeth makes quick work any size stump. Left behind will be all the grindings like a compost pile.
Sunset Tree recommends stump grinding because it will be difficult to replant in the same area again. Tree roots usually extend out into the yard almost as far as the branches did on the live tree. By leaving the tree stump in place, the thicker roots are also left behind which makes it impossible to plant a new, replacement tree anywhere within 10 feet of the old site. If you want to replace that old shade tree with a new one, then the stump must be ground up in order to create some space.
Also old tree roots take forever to break down. Once a stump has been ground up, the roots break down quickly as well. Roots attached to a stump seem to last much longer and can get in the way of planting shrubs, flowers, and other landscaping amenities.
Certain trees do not die when they've been chopped down, and will send up suckers the following year. These suckers are a bit like the multi headed Greek "Hydra" and instead of coming back as one tree, will return as 10 or 20. Grinding down the stump and the thickest parts of the root stump prevents the tree from turning into an unwelcome addition to your yard.