Good Morning, 

Thank you for contacting the U of M Extension Service. 

I agree, you don't want to be cutting or digging out the tree roots as it would cause tree damage. 

Please click on the following link about surface tree roots: https://hort.extension.wisc.edu/2024/05/28/how-to-deal-with-surface-tree-roots/#:~:text=To%20protect%20the%20tree's%20roots,in%20between%20the%20tree's%20roots.

Here is another link about watering trees that discusses mulching at the bottom: https://extension.umn.edu/planting-and-growing-guides/watering-established-trees-and-shrubs

50% of the tree roots are within a foot of the top of the soil. You did not indicate how large your yard is or how large the tree is. If you planted low growing plants in between the roots it would create "soft landings" for bees and caterpillars. They need the leaves and plant cover to live. Our turf grass lawns that are mowed and covered with pesticides are a food desert for nature and an ecological trap. 

Here is another link about soft landings under trees. You might want to consider this to help our pollinators thrive: https://www.pollinatorsnativeplants.com/softlandings.html#:~:text=These%20plants%20include%20Solomon's%20Seal,attracting%20a%20variety%20of%20pollinators.

If you scroll down, there is a list of plants you could include in a soft landing. 

Good Luck!