I agree that a hot horse like that will probably never be a solid, steady trail horse and will require a rider who can deal with histrionics. So she may not, as you say, be that happy hacker who suits you. But it doesn't mean being exposed to it won't do her good. After all, the way to give her confidence in herself and her rider is expose her to scary things so she learns they are not scary. It just might take the most patient rider in the world with a velcro butt, who isn't phased by equine meltdowns, and who is willing to take as much time as it takes to do it.
If she will go on the trail and be kind of fine going behind another horse, then do that. It took one of my horses about three years of lots and lots of trail riding following other horses before she would take the lead willingly.
I think it's a good idea to expose her to an obstacle course, to a running car, in the arena or out of it. I doubt it will make her anxious about the arena, and even if she is going to be a dressage horse, she needs to be able to deal with stuff. Horse shows are not safe, controlled environments at all! There are flower boxes, signs, banners, flapping tarps, cars, people zipping about on quad bikes and golf carts, other horses going nuts, and more.
That's just my philosophy. Expose a horse to anything and everything. I've never met anything so hot that it didn't chill out even a wee bit after lots of desensitization work, unless it had some other underlying health issue. And it might take a lot of work. As an ID-Connemara X, however, she should have some sense buried in there.