I don't have surface tree roots out in the yard but when it becomes dry in summer, the feeler roots will search for moisture and nutrients and manage to get into my raised beds. They will travel a good distance.
I only wished my raised beds were spaced further apart for working between them. I can get a wheel barrow between them but if I want to kneel between them, I have to turn sideways. Big feet I guess.
I can't say I know much about growing grass. I had my house remodeled three years ago and am patiently awaiting a circle drive around it. It is too muddy for them to work right now but the drive will eliminate a good deal of mowing for me.
When that is done, I hope to do the landscaping inside the circle drive which will be centipede grass. That type of grass doesn't do well in all soils but it does here and I like it as it doesn't require extensive mowing. Currently I use a 20" rechargeable mower for close work and a tractor w/bush hog for the rest. End of February and many here are already mowing. The first growth is weedy. If it weren't so wet, I'd do mine. I assure you, by the end of summer, mowing gets old.
I don't think it will hurt to trim back some of those surface roots. I'm curious if you have lots of shade? That may affect how well your grass lawn will grow. If you don't have lots of trees, you might consider plantings around the base of the tree so the roots aren't an issue. Just a thought.
BTW, thanks for pelting me with acorns. Appreciated but not necessary.