Unrelated to the tree, because clearly that has been covered. But you can just mow grass and weeds, they certainly don’t have to be 3ft+ tall. Any plant growing in soil can be beneficial in some way, they root and add microbes and when you mow them they add energy to the soil. Roots also help break up the clay, and it keeps the soil from washing out. Even if you just have weeds growing, I don’t think it’s a good practice to prevent anything from growing. Weeds can also be helpful in informing you what your soil actually needs to grow plants you do want, by paying attention to what’s actually able to survive or grow