...what can I, the human, learn from normal arena-riding teaching that would apply to trails? Of course I mean the level of instruction that's aimed at getting the student to compete in shows. Obviously you could teach basic riding in an arena (if you have one), probably better than on the trail...
You have largely answered your own question. I started riding at 50, and had a back injury within months - a back injury that is only now clearing up after 4.5 years. 40 years of daily jogging left me tight in all the places that need to be loose for good riding. I had ZERO experience in reading a horse or anticipating their reactions.
There are tons of things I can learn safely (at least more so) in an arena that apply to trails:
Feeling fully relaxed and comfortable with a cantering/galloping horse.
Staying on a horse that is jumping, particularly when no one but the horse sees what it is she is jumping...
Learning to read when she is about to bolt, and how to stop her before we're flying down a trail she isn't balanced enough to handle.
Becoming comfortable staying on a horse who is spinning around or jumping sideways.
I wasn't born on a horse. I wasn't born graceful. Trail rides here mean going between cactus. Ever known anyone who went thru a cholla cactus? I have. Not me, happily, but a friend. And that was on a bicycle, which had no means to drag him for another 2 miles...
My horse is no more graceful than I am, Maybe even less. Unlike her, I've never fallen on my own while running. I'm convinced she had never been out of a corral. I've seen her lose her footing while cantering without a rider in an arena. If she did that in a wash around here, the rocks under the sand could break my back. It is a fun place to ride, but not on a horse who is a klutz.
BTW - in a couple hundred yards, that wash gets very narrow and rocky. If she bolted mindlessly at that point, my choices would be to stop her within a couple hundred yards or have both of us get hurt. That has required training for her, and training for me. There is a place to turn out of the wash, but she would never hack that turn at speed.
I doubt upper level training in WP, dressage, reining etc has direct application to trail riding. Trail riding has its own challenges. Those are sports people do in an arena because they find them fun. Some horses love it too. To the extent those sports teach balance for both horse and rider, and confidence and how your balance helps or hinders your horse and confidence you can stick to your horse no matter what, they help. But it is also OK for people to enjoy those activities for their own sake.