I need to cut a 6-9" deep drainage in a yard. Approx 25 yards long. Anywhere from 24-36" wide. What piece of equipment do I need?

Will the little stand on skid steers have the power to cut untouched ground or will they just spin?

I rented a " medium sized one from Home Depot and it worked fine. Put in a river rock drain.

75 ft runs needs ~ 18.75 inches of drop for draining water unless the swale is cement

You don't have a neighbor this is pointed at do you?

And are you talking about a ditch witch, large walk behind chainsaw looking thing? Yea that'll work.

If there is grass, it may struggle to bite into the surface and just slide along the grass. If you can till the grass surface or cut and remove the sod, then yes, a mini skid will be plenty to remove that type of soil. There are different techniques to get a gentle slope to the center, mostly involving cutting in a narrower trench at the centerline at the desired depth, then working to scoop out or doze out the sides to that target depth on a slope.