I know you've already heard this, but stop being so hard on yourself about this. Just let your instructor know that you are not ready for trail riding yet, and for now you would like to stay in the arena. Just until you feel a more relaxed and comfortable on the horse. You should be able to walk, trot, and canter in the arena without feeling that you are losing control before you attempt to go out on the trails with anyone. And when you do go out on the trails, at first it needs to be with people that are considerate of your experience level, friends that will walk until you are ready to trot or whatever else. When you do feel ready for trail riding again, try to push yourself a little out of your comfort zone, by trotting a few steps, and build gradually on that.
There is nothing wrong with starting slow, especially since you have a particularly nasty incident in your past. That in and of itself would stop a lot of people from ever trying again. My mother had a childhood riding accident that broke her arm, as a result she is terrified of horses. She will only pet one if it will stand absolutely statue still. So see, you are doing really, really well. Just keep trying, and let your instructor know how you feel about doing what it is that they ask of you. If you are really scared, tell her. If it only makes you a tiny bit uncomfortable, try it. You have to keep pushing your comfort zone to get just a little bit bigger.
As for the kids doing better. Being as this has been a kids horse for years, it is likely used to the antics of children, and behaving properly. I once had a barrel horse, and a good one too, that was a handful for adults, but one of the best baby sitters in the state. I could literally come out from running barrels, and but a little kid on him, and he would plod off and never take a side ways step. The kid could even ride him into the alleyway and walk or trot barrels on him. He loved kids and would never do anything to upset one, he would cart them around for however long they wanted to keep riding. He never offered to misbehave with a kid on his back, but was always testing older people. Basically to ride him you either had to be a little kid (less than 10 years old) or very experienced, go figure.
Keep trying and stay safe.