Here in NC, local round bales are cow quality usually..and more likely to have weeds you don't want for horses. Mine were fed on round bales..looked like straw. When we got them we changed their feed, and fretted for months over the foals that were due. All was fine, and milk came in, but they will over eat the higher quality quickly after needing to take in so much low quality hay.
Good round bales can be found, but are hard to store, move, meter out..and transport without a tractor and pole. smaller ones do fit in a truck bed, but can't be moved without grief and help. They should be stored on their sides to shed water..and they can roll..are dangerous around small kids and small horses.as they can be rolled on . Stored flat, the water can get in and mold the inside..not a problem for cows really, but deadly for equine.
I buy the best square bales I can find, when I can find them..and my farmers keep passing away. Breaks my heart. Most of the younger growers are baling for their cattle, and horse people are buying feed. I have gone to the bagged timothy cubes mostly, easy to know what I have, and to store..with the hay reserved for slow feeder bags in each stall. It's been really pricy.
Horse hay used to be $3. Last couple of years it got up to $15. Or more. Many sold their hay out of state and prices went crazy. I bought 200 bales last summer for $3.50 and was quite pleased. I am waiting for the second cutting now, and wondering about the price! Craigs List has been a good source..but you have to know what you are looking at. Many people here won't tell you who they get their horse hay from..because you might buy out from under them. Sad, but true.