The horses etc would not be a problem for the trees, just add fertility!
It could be a problem if you were going to grow any edibles there - hence the reference to E. coli.
The idea of the mushrooms is that they can remove contaminants, even the E. coli. From the abstract: "process known as mycofiltration, is a promising new technology for enhancing biofiltration of stormwater, graywater, and agricultural runoff
myco-filtration would be filtration by mushrooms and agricultural runoff would be the drainage from your horse pastures. So if you wanted that all cleaned up, growing mushrooms there could help. But as long as you aren't growing any edibles, you really don't need to worry about it.
So yes, the willows or other water loving trees would like your spot. Sounds like they would be fine there, since there's no pipes nearby. And yes, they would soak up a lot of the water, though as in my Quaker Lake, probably not enough to make it dry.
I am always opposed to mono-cultures - more vulnerable to any pest or disease that comes along and less diversity of habitat value. Perhaps think about having a few different kinds of trees, instead of a half dozen willows?
swamp maple, catalpa, serviceberry, hawthorn, water tupelo, eastern cottonwood, water oak, swamp spanish oak, black willow
These are all native trees for you, that are water loving, that have lots of habitat value for birds, bees ("tupelo honey"
), beneficial insects, butterflies, etc. There are over 40 species of birds that love the service berries. Often you can spot rivers from a ways off, by the line of cottonwoods growing along them. Birds use it for nesting material and eat the seeds and it is a larval host for Mourning Cloak, Red-spotted Purple, Viceroy & Tiger Swallowtail butterfies.