ROFL, too funny! Maybe I should go see if any of mine have died yet? They never had anything but big round bales that have been stored outside! Shoot, my oldest is 11 next month, that's 11 YEARS of eating cow hay and never a sick day in his life. My new prego mare is happily munching on a big round bale right now, she actually PREFERS it over a small bale of straight alfalfa hay that has been stored inside.
We too feed round bales that are stored outside. Our's is wheat hay baled for horses. They will jump over alfalfa to get to the wheat hay. Only drawback we can see is keeping the weight off of easy keepers. The three horses eat about one bale every 2 weeks. It's pretty dry here so an open bale lasts well. If we do get rain the horses just pull off the stuff they don't want and sleep in it!
We've been feeding round bales for over 12 years without a problem.
Provided they are stacked properly and are not more than a year old there shouldn't be a problem.
Most of the hay here is stored outside , barn storage is rare . I have had outside stored bales over the last 5 years . You do have to remove the outside edge but after the weathered hay has been removed the hay inside is the same as barn stored hay.
They also store straw and lucerne ( alfalfa ) outside with no problems