I used to feed round bales under shelters. We just feed round bales in winter. I got tired of cleaning out the run-ins. Now I go set a bale on a poor spot in the pasture-we still have some bare spots. I may feed two in a row in the same spot. I set the bale where I want in, and go get the ring from the last spot with the spike on the loader, and set it over the bale at the new spot. They won't go bad in cold weather before they get eaten, and I don't even worry about a rain or two if it's cold.
If it's going to rain for days, I'll feed them under the roofs.
In spring, I throw some grass seed out on the leftovers around where the rings were, and spread a thin layer of dirt over the seeds. The horses don't bother with that grass until probably next year, because they have the rest of the pasture coming in at about the same time.
Hay handling becomes a ten minute thing (have to drive the tractor a few hundred yards to the hay shed) once every 5 days or so with this method, and cleanup has gotten to be very little time spent.
I do keep the rake on the back of the tractor in winter and keep the rest of the pastures cleaned up with it, and use it to clean out under the shelters when needed. The stuff I rake up goes on the years composting pile. I wouldn't want to do it any other way.
With a loader, hay shed, and rings total hay handling requires very little effort. Unload a trailer of squares by hand, and a load of rounds with the loader in ten minutes, and tell me which you prefer.
I feed my round bale on a pallet with a ring. And yes, I have a cattle ring. I've used it for years but my horses are not idiots. I'm getting a new ring (maybe tomorrow) because my current one is getting old and pitiful. I'll get a "horse" one this time. I love feeding round bales because I know they always have something to eat. My three will go through a 4x5 bale in about a week so no mold.