So first thing, her neck, something typical of a horse that is not "working" over her neck, just flexing at the poll and carrying on, which is pretty obvious with her unsteady contact and lack of proper self carriage in the video. I don't know how it happened but with her high neck set and uphill build (although if you look at her "main mass" it is angled downward because she lacks a solid top line over her withers and back) they've managed to make her ride downhill and yet still need to stretch down (i.e.- walk and all nose out and up stuff) which to me screams that she's being ridden incorrectly.
I also see what Kayty is talking about but I see it somewhat consistently, this could again be "rider error", in which the rider is riding unevenly and making the horse unbalanced (which would be most apparent during a downward transition) or the horse could have a hind end issue. This horse is not using her hind end when she moves, which could easily cause back and upper hind joint problems (stifle and hip). She also has a mildly long back, which makes it weaker and not being ridden over her back would make her weak in her back, top line, and hind end (which will bring about a lot of injuries because they just can't work that way effectively, especially jumping).
I'd suspect she chucks herself over jumps because she has no muscle. At 9 years old she should be a third larger than that, she looks waspish and under developed. Look at the size of her head, bone structure and joints, they're not as dainty as she is. When I looked at her pictures (admitedly before I read anything) I assumed she was 4 or 5 judging by her structure and muscle build. But before I obsess over her lack of muscle, if she got regular work making her use her hind end and build the strength and then was taught that she could then power from behind over the jump rather than chuck herself over because she doesn't have muscle she might develop a better jump. But again, the chucking could be due to pain or injury as well. I'd suppose, though, that the chucking is due to not really knowing what she is doing and not having the proper muscles to do it correctly. I've seen a lot of horses chuck themselves over jumps and then you get on and give them the right skill set to do it right and suddenly they are like "woah I can use my hind end, this is so much easier!!!"
As for hunters, I doubt she'll do very well, between her knee action and her chucking herself over fences (can't really judge how her jump will be at the moment), she won't place and will make hunter/jumper shows a waste of your time unless you stick to the jumper ring. Although, my advice would be not to even jump her until she knows how to use her hind end and builds some muscle so she doesn't potentially hurt herself. She might make a decent eventer, I doubt she'll be going Advanced any time soon but Novice and possibly Training isn't out of her spectrum.
I will encourage a thorough PPE with x-rays. I always at least do x-rays of the front hooves and hocks, and if there is an "iffy" area I'll x-ray that as well just so that I know what I'm working with and what sort of things might be a red flag upon resale. Even if you think that you'll never ever ever never ever sell her you always want to know that you're not buying a horse with problems, and in the event that you had some horrible financial situation arise you could sell the horse if need-be.