I can see pine working if you live in one of the drier parts of the country.
Everywhere else, hardwood (like oak) is going to have a much longer lifespan than softwood (pine, poplar, etc). Of course, it also comes with a higher price tag.
My run-in shed has treated structural elements, and the outside is rough-sawn oak applied as board-and-batten and thoroughly stained. My dad (who built this one) has a similar building that is over 30 years old and still in perfect condition.
If you have 'beaver' horses, you can screw cheap 2x4s to their favorite chewing spots and just replace them periodically.
I have a run-in shed made of hardi board 4 x 8 foot panels. My two year old colt has decided to chip away at the bottom of one panel and has broken off a large section, about 2 ft square. He did not like the taste of it, and it took him all summer to do this. He also ran into the siding and managed to acquire a soft tissue injury in his shoulder. Left a rather large dent in the shed. Yes, horses can damage hardi board, but at least he is not eating it.
Horses think pine and poplar are tasty. I use rough-cut oak or (if it has to be pretty) Southern Yellow Pine aka SYP. SYP is available at regular lumberyards and used mostly for stuff like floor joists. If you're in the Northeast, rough-cut Hemlock will work.