How tall of a shade tree are you looking for? American persimmons are “self-pruning” - i.e., they drop a lot of smaller branches. Also, the fruit falling to the ground can be messy - especially if you let the tree get full size. The soft fruit will often split when it hits the ground from high up. I have large wild persimmon trees on my land, but I prune my grafted varieties to keep the fruit within reach from the ground or harvestable with a step-ladder. American persimmons also put up root suckers quite far from the tree, which are easy to mow but might be annoying if you have garden beds or beds near your house that you don’t want to weed.
Mulberries are much faster growing, but they can also be messy trees when the fruit falls on the ground. My mulberries attract a lot of birds and the ground under them has a lot of bird droppings when the fruit is ripe. Mulberries with purple fruit will stain driveways and walkways. Mine also tend to have broken branches whenever there’s a storm. But if you want an attractive fast growing tree with edible fruit and are willing to put up with these drawbacks, they are hard to beat.
Some of the astringent kaki persimmon varieties are large enough to be smaller shade trees (20’ to 30’ feet? - I’m not sure how tall they get if you didn’t prune them for height). Some also have attractive red fall foliage like Tecumseh and Great Wall. I grow Tecumseh, and it’s a gorgeous and productive tree. But mine is grafted to an American persimmon rootstock and produces suckers. I think you would probably want to prune a young kaki tree to get a good structure as a shade tree - many kaki persimmons I see in my area have heavy branches drooping down because they are loaded with fruit and propped up with stakes to keep them from breaking or laying on the ground. This can be avoided with some late winter pruning.
I’m also very fond of my Japanese Raisin Tree (Hovenia dulcis) as an attractive shade tree. The “fruit” is more of a novelty, but the small flowers attract a lot of pollinators. It also “self prunes” and drops small branches, but it hasn’t suckered at all for me.