With broke horses, I take the snort and poop as a sign they’ve warmed up and physically relaxed. There’s a spot on our home trails about 15 to 20 minutes from the barn and it’s clear a lot of horses poop there! I think they also poop when they see other horses poop. Of course there is also nervous pooping, even our good loading horses poop more when they see the trailer and in the trailer
It’s been my self-board conclusion, cleaning stalls at different times of the day or repicking to tidy before I leave, that horses poop about once every two hours, and if they poop five times in the trailer loading and trip, there will be a bit less poop in the paddock the next day.
It’s not much fun to clean up cross ties, but also not much fun to pick out the arena after you ride so 6 of one, half a dozen of the other.
My mare remains reluctant to poop while she is moving. So if your young horse is working nicely for you in the arena she might not feel she is allowed to stop and poop (which is wonderful, the badly timed poop stop can be very embarrassing in any discipline!) and hasn’t figured out poop and walk yet.
It’s possible that longer trail rides will help her figure this out, especially if she sees another horse poop or sees fresh poop on the ground. Or gets to relax more.
My mare will also often go out to her paddock and take a good long pee when I arrive and and she knows we are going riding. After a trail ride she won’t pee tied to the trailer on a gravel parking lot, but will once I take her for a grass walk. She has peed under saddle on a longer trail ride once or maybe twice in ten years.
Anyhow, for now I expect that your young horse just isn’t fully relaxed under saddle, but isn’t so tense you are getting panic poop. Give it time. Not pooping in the ring isn’t the worst quirk out there!