Grasses can, indeed be a problem under trees.
They are horrible
water hogs.
Many grasses form a mat-like root structure which allows very little water to penetrate below it.
This not only robs the tree's roots, but also the deeper soil microbes, which the tree needs for optimal health.
Individual plants under the tree's
canopy shouldn't create this problem. As their roots expand outward and down, they will learn to co-mingle with the tree's roots, each of them assisting each other by exchanging nutrients they have in abundance for those they lack.
Just be cautious if digging holes for transplanting larger plants. Don't go in there and begin digging 1' X 1' holes everywhere. I wouldn't do a lot of planting shortly after pruning - if you prune 10% of a tree's branches, it will self-prune 10% of its roots - not a good time to go in there and damage more.
Tree roots are dynamic - except while dormant, every day some will die, and new ones will form. So even if you damage/kill a small proportion of them, the tree
should continue as if nothing had happened.
With a wide variety of plants growing around your trees you will be attracting a wider variety of beneficial insects (including pollinators) and also a wider variety of soil microbes which bring life to your soil - a Win/Win situation.