I would make small terraces over the eroded areas by the barn. As mentioned, wood edging are not hard to put in, then “I” would fill the terraces with rough edged stone or crushed concrete. Tamp down the fill to make a harder surface that will stay put. The rough edged stuff will lock together with use, stay put when the water goes thru or over it.
We have a solid, crushed concrete driveway to the barn now. Just kept adding more loads after the first load proved to be so good, despite rushing water trying to move it. Truly, the more you drive on the concrete, the more solid that driveway gets. Took a couple years to get in enough fill, but it is now a raised drive, above the paddock dirt beside it by about 6 inches. That is enough to contain flooding in the paddock when we get heavy rain, so water goes down the dug ditch into the big drain culvert pipe under the driveway, not into the barnyard or backyard anymore. That paddock has dug drainage ditches on two sides, fed by water from across the road drain (6") tile, via the under the road culvert, along with the swale drain pipe (6"). We ARE the low spot! There can be a LOT of water being pushed downhill to us.
In the past we had done sand, gravel, river (smooth sides up to cobble sized) stones. They all were inadequate for the job, washed out, away. The use of crushed concrete was suggested by a contractor and was cheaper than limestone. It has been a great material for us! Over the last few years we buy a load and spread it around on places that make us say bad words! Gates, route behind the barns driven daily to clean stalls. A standing pad for horses to get up out of the mud that usually lasts months.
The concrete can be crushed in various sizes, larger or small, it stays in place pretty well to be packed down. No tender footed horses here, bare or shod, they stride right across it going out or coming in.
The animals grazing the land previously could only benefit soil with their manure. Sheep is better fertilizer than horse. I would get soil tests done so fertilizer you spread will add what is needed for good pasture. Use Ammonium Sulphate instead of Urea, it is a better Nitrogen source with less issues. You WILL have to ask for it, they automatically give Urea to buyers. Benefits are MUCH less volitility, won’t evaporate away if not rained in immediately. No chance of Urea Poisoning if you don’t use Urea! Equally beneficial to the plants, it costs about the same as Urea. My husband the Farrier has seen cases of Urea poisoning, very ugly when the hoofs sluff off! No fixing that. Doesn’t happen every time Urea is used, but we are unwilling to take any chances. Urea Poisoning can affect any hooved animals, not just horses.
If you want to lay tile, get the stuff with th knit covering, called a sock on perforated pipe. Sock keeps dirt out of the pipe MUCH better than using plain perforated pipe for tiling to help control water flow, drainage of wet spots. We have a lot of socked pipe here, makes a world of difference when it is wet outside, much faster drying of The Mud we get!