she looks adorable. The first thing I see is that she's a little low with her knees and as for you - you tend to lie on her neck even over smaller fences. Your weight is not in your heel and your lower back is hollow (on the flat).
I'd work on a lot of two point and bounces and gymnastics to get your pony a little sharper and get your weight more in your heel. I'd also do some work w/o reins as well as with your eyes closed over fences (you can do this on the longe with a good trainer). This will keep you from lying on your horse in the air.
I'd really like to see more weight into your heel and a better base of support - which will help your upper body not lean on your horse. In jumpers it's not about being fast, it's about being tight and clean. Tight turns require a VERY strong base of support, with angles relaxed and flexed, heels down, and weight in your lower leg wrapping around your horse.
What I see in the video is a horse on the fore lacking impulsion and bend and a rider gripping with leg and steering with reins. In jumpers your horse MUST have impulsion and be off the fore, with a relaxed poll, engaged back, and powerful hind. Regardless of her trim, the standing martingale (illegal in jumpers) implies that your horse needs help holding her head down which tells me she is not engaged nor does she know how to properly engage her hind end. A horse moving with impulsion does not need a tie down or standing martingale. As for your balance, your turns show you leaning WITH your horse. To make a proper rollback or even a regular turn on course, your weight should be in your outside stirrup, to free up your horse's inside hind to reach underneath herself and bend through the back and ribs and moving through the shoulder.
I'd also guess that headset has nothing to do with why you don't always place in hunters. Hunters are able rhythm, balance, style, and consistency. To have this again the horse needs impulsion, though with less roundness through the back than a jumper. Judges look for bend through the neck,shoulders, and ribs, and relaxation through the poll and jaw. These are things that I do not see in your videos and things that are NOT related to her trimming.
Before progressing to jumpers, i'd take some dressage lessons on the flat for the benefit of your horse's fitness and your ability to properly use your outside aids and legs to steer and use much less hand and get your horse off the forehand. Then for your position i'd do what I recommended above working on two point, steering without reins, strengthening the lower leg, and learning how to use your thigh and seat for bend and steering independent of your lower leg (calf/heel) and upper body/reins.
Until your leg is much more secure and your upper body independent, i'd also recommend losing the spurs please! Spurs are NOT for go. Spurs when properly used are to fine-tune the aids we give to our horse such as take off here, jump long, jump big, jump round, jump wide, etc.
One last piece of food for thought:
Jumpers is known for off-distance fences and various fence styles that you do not encounter in hunters. I always ask my students the following before allowing them to do jumpers:
Do you know how to ride a 3 3/4 stride line? What about if that is followed by an oxer 6 strides away or what if it's followed by a tight in and out or a long in and out? How does that change how you ride that first line? Do you know the difference in riding, approaching, takeoff, and landing a vertical v. A triple bar v. A fan v. An oxer v. A liverpool? How does one of these on course affect the preceeding and following fences and change your ride? How does one of these in a combination change how you approach the combination? What's the difference in how you ride an in and out that is 2 verticals v. An oxer to a vert, v. A vert to an oxer?
ALL of these are things that you need to know and understand and be able to navigate AT SPEED to successfully do jumpers. Are all of these elements in the lower levels? No absolutely not. But they ARE in there - sometimes just one or two of those elements but they will be there. Rollbacks in comparison are the easy part of the course!
That said - please don't become one of those riders that uses raw talent and a good natured horse and a set of spurs to run around a course on the fore steering with the bit. You definitely have tons of potential and i'd love to see you reach it!
Best of luck to you!