I am going to address just the changing speed, on a loose rein, versus any attempt to run towards the in gate, as I see perhaps a two fold problem.
I think by what the Op posted, she wishes to be able to rate her horse, on a loose rein, and assume she is riding along the rail
That is a final skill, once the horse is very correct , far as body control, using contact as needed, while driving with legs, working off the rail. The horse has then learned to stay between legs and reins, thus had guide. You have also given the horse chances to stay correct, on a loose rein, picking up contact again, if needed, in combo with legs, to fix the horse. Rinse and repeat
Then, when you are ready to ride on that rail, you try not to school, making the rail a good place to be. You thus trust the horse to go at the speed you want, on a loose rein.
If he does not do so, take him off the rial and school. There are many ways to slow a horse without bit contact,, that you use, in trianing, so that when you do show that horse, you have those 'buttons' to slow the hrose, without rein contact.
The hrose has to learn that leg contact does not mean just to speed up, but rather to drive deeper, which in turn, will be a cue for the hrose to rate speed
While you are training, if the hrose speeds up, add more pressure to your stirrups, as the initial cue, then stop the horse, back him up, and then give him achance to go correctly on that loose rein
Soon, the horse will learn to slow, with that added pressure and leg contact alone, knowing if he does not do so, he will be stopped and backed fairly hard
This then leads to being able to rate that hrose , in the show ring, eventually, off of leg and seat alone
If you want to ride a horse eventually on a loose rein, then you have to correct him, when he is ridden in that manner, without resorting to the bit