Hum, Utah's mascot is indeed "Swoop" the red-tailed hawk. Not sure red tails are big enough to actually take out a duck, but the South Dakota State Jack Rabbit is quaking.
The Ute was even more formidable than I thought. They dominated the West, and when they got horses in trading with the Spanish they expanded, the Comanche (which is an Anglicization of the Ute word meaning "enemy") were Utes who moved to the Plains. They raided other tribes to capture women and children which they traded with the Spanish for wares and weapons.
As for their power as a Mascot, this picture of a Buffalo on a beaver skin tells you all you need to know about how the other mascots would fare against these skilled trappers and hunters.
Bronze armor would be ineffective against the weapons of the Utes, both rifle shot and the bows they had would have penetrated the armor., as would steel axes they got in trading with the Spanish and Americans. Also the stirrup meant they could shoot arrows accurate from horseback. But they probably had to dismount to fire rifles.
But that is irrelevant as Swoop the red tail hawk is the mascot. He'd be no more effective against Trojan armor than the ducks.
But Utah instead went with a PC mascot, rather than their namesake, so they are pretty far down the mascot list.
The classical era Greek warriors, which the Trojans were part of, only had chariots. The stirrups had not yet been invented, so effective mounted combat was still hundreds of years in the future. The were basically infantry, with Javelins and Bow and Arrow support. But they wore extensive armor, from head to toe, making bow and arrows pretty much useless until centuries later. Achilles getting hit in a spot where the armor doesn't cover is what that story is about. Look at the armor here:
They probably had arm sleeves like they have leg sleeves, secured by straps. The foot covering surprised me. But obviously the heel is not covered as you need to move.