Willow trees: will grass still grown underneath

[Amaggio]

Hi there, I'm planning my chicken run/goat pens and need a good shade tree that i can also occasionally feed to my goats. I was looking at the weeping willow tree because it's edible and gives good shade. However, i have no experience with willows at all. It seems they would like my clay soil but will it give too much shade that plants/grass won't grow around it? They can get pretty big too, would dwarfing it be an option while maintaining a fair amount of shade?

If someone has a better suggestion for a shade tree that would be compatable with my goats I am willing to receive any advice.
Thank you in advance.