Realistically, the Mustang Heritage Foundation knows what they are doing when they assign the horses for these events. The horses are generally the older, plainer horses that likely wouldn't have been adopted on their own. Therefore showcasing them through a trainer's challenge helps these horses find homes instead of going to permanent holding facilities.
With that being said, typically, the longer a horse has been wild, the more difficult they are to domesticate. While older Mustangs can be gentled, they generally take more time because they have grown wiser and more committed to self-preservation.
The differences between mares and geldings tend to be more universal across breeds, however. This is because their social differences are deep rooted in their herd dynamics in the wild. In nature, mares form stronger bonds with their herds. Stallions in the wild are more transient. They are born into a family band, get kicked out when they become too old, join a bachelor band, try to sneak into whatever herd they can make in into until they establish their own herd. But then stallions get too old and they return to bachelor bands once they can't maintain their mares. On the contrary, it is common for mares to remain in the same herd for the majority of their lives.
Because of this, mares tend to either form a very strong bond with their owners, or the opposite can occur. My favorite comparison is that a mare will either work twice as hard for you or against you. The way mares bond is a bit of a double-edged sword in the sense that when they give you their heart, the relationship can be much more rewarding. But if you give your mare a reason to distrust you, she'll hold it against you much longer than a gelding. Overall, geldings tend to be more forgiving than mares.
Regarding the hormonal influence to mares' behavior, there is such a discrepancy that you can't really judge a mare's behavior based on their hormones unless she is pregnant or has a foal at her side. Just as some women's behavior isn't significantly altered on a monthly basis, I've had some mares that I couldn't tell when they were in heat! But other mares have been so significantly affected by their hormones that they would come into heat instantly when introduced to a new gelding. In the wild, the lead mare gets bred first. Therefore if you have a mare that is more concerned about being bred than about what you're asking of her, there is almost always an issue with leadership.
If you have a mare that is extremely influenced by her hormones, consider spaying her to remove hormones from the equation. A spayed mare acts more like a gelding because there are no reproductive hormones muddling her thoughts. Spaying is a quick and easy procedure that many people don't know is an option!
Overall, geldings tend to be easier because they tend to be curious, playful, and forgiving. Mares should not be counted out, however. Mares simply require a greater level of horsemanship in order to access the strong bond they are capable of.