The horse I’m currently riding was being leased in-barn, but after being caught mistreating the horse on more than one occasion, the lease was terminated and she was kicked out of the barn.
My trainer started using him for lessons since she was just getting back into giving more lessons and he was used for a variety of riders. After his trial as a lesson horse, it was decided that he just wasn’t going to work out. He’s broke, but hasn’t done a lot of work very consistently throughout the years, so he just wasn’t broke enough or a simple enough ride to use consistently as a lesson horse. I started riding him in October and at the time one other adult was taking a weekly lesson on him. Around the 1st of the year, he wasn’t used for lessons for a couple weeks, but due to a lack of lesson horses, he started being used again. Now the other adult is leasing another horse in the barn, so the horse I ride is no longer used for lessons at all. Other than by me.
I hadn’t jumped in about 6 years before I started riding this horse, so I was definitely rusty, but nothing he did was ever anything I couldn’t handle. It was just a bit longer process to get things correct due to fitness.
I’ve ridden several other green lesson horses over the years and I’ve never had an issue. But none of them are what I would call “problem” horses, they just needed more miles.
In your case, it sounds like that horse needs more training rides before he’s going to be comfortable with other people riding him. If he ever is. If I were your trainer, I’d limit the rides by others on that horse and also limit who is riding him to riders who can handle him. If you’re not comfortable riding the horse (and I wouldn’t be), I think you need to tell her that.