Mike,

Unfortunately, there is no easy solution to surface tree roots. If you do apply topsoil, do not apply more than about 1/2 an inch. If you apply more, there is a chance that the surface roots can be negatively affected. Surface roots are important for air exchange and applying too much soil can "suffocate" them. There is also a chance that the roots can emerge again even after applying a thin layer of soil. Grass and trees rarely get along---they compete intensely for water/nutrients---and even if you use shade tolerant seed it can still be a challenging environment for them to fill in these spots.

Option number two involves making a large diameter wood mulch ring around the tree, covering the surface roots. You can go deeper than topsoil in that case, 2-3 inches, but that would kill the patchy grass that remains. You also would need to periodically add new mulch as it breaks down. However, trees (even mature ones) typically do better with plant-based mulch. When those wood chips break down, it can also increase soil health and help with compaction. Some folks also plant broadleaf, shade plants in the mulch ring (like hostas) which are less competitive than grass.

Removing the surface roots (grinding/cutting) is probably the least good option out of the three. It can be hard to anticipate the effects of doing so without the help of an arborist or tree professional. The roots could eventually come back or removing them can compromise the tree's health.

Here are some articles about surface root problems/solutions:

1. University of Wisconsin

2. University of Kentucky

3. Purdue University

Hopefully these options help,

Shane