You shouldn't be on her neck if you're jumping and you know she has a tendency to stop... You shouldn't be leaning on her neck at all, ever. Have you ever taken a jumping lesson from a trainer? They would have shown you the 3-point seat, and how to stay balanced without leaning on your horse's neck.
If you're throwing yourself at the jump, instead of waiting, any horse will stop. Because you say she starts the jump, it's not because she's not willing... it sounds to me, based on the way you've described the situation, that you're not staying back and out of her way so that she can take care of her job.
I rode a horse that would slam on the brakes if I even leaned slightly too much before a fence. I would go back to tiny crossrails, in the arena, and see if you can't get someone to watch you. Ask them to see if you seem to throw yourself at the jump. You have to keep your eyes UP, shoulders back but not tight, and stay relaxed. Let the horse move beneath you.
Pushing is ok, but not when it's a sudden, three strides before the fence, WHAM type of push. You have to be consistent. If you're going to push and encourage, do it all the time, not just when you think she's going to stop right before the fence. When you're approaching the jump, you shouldn't have to change anything about your position, energy, or gait. That'll mess your horse up even more than just throwing yourself at the fence would.
You have to wait, wait, wait, you can't expect your horse to jump when you're not staying out of her way and allowing her to do what she's supposed to do. If you need to encourage, be consistent. Don't wait for her to stop to push and get her energy up. You can't just meander up to a fence. The horse needs to have energy and power, not speed, but power. I rode a 14hh pony when I was younger that could trot a 2'9" fence, because she had power and energy. The moment you stopped letting her do her job, you'd find yourself on the ground or with a refusal.
A video would be great, if you could get one. Then we could see what's going on, maybe you're missing something in your descriptions. Good luck!