I had a TWH that was a hard core pacer. She was lprogrammed pacer (but I beleive was a stacked Bick Lick at one time due to some suspicisous scaring and many former Big Lickers tend to be pacers once the stacks come off) Since this was her most comfortable/chosen gait (and she did not trot, rack, amble or RW even in liberty) I chose to work with this gait. The strug out pace is not comfortable to ride. I taught her to "jog" pace (pacing slow and rythymicly like a WP jog) very nice.]
Some ppl cannot tell the difference between a stepping pace and a rack. Some ppl cannot tell the difference between a slow rack and a run walk though the timing of foot falls are a bit different. One has a larger over reach (over stride) than the other. Did this teen actually ride this horse infront of you showing that this horse could actually perform these gaits and actually know the difference herself. I find that most rackers don't RW as well as they rack and vise versa. Some ppl don't even know the difference betwen a rack and running walk. Yes there is a difference.
If this horse performs other gaits at liberty then it very well may be the way he is traveling. Hollow backed, stiff neck, lots of tension (which could be rider related) and hind legs that tend to rail behind him instead of underneath him. Conformation plays a roll in this also. Long back, short upright neck, does he stand camped out or long in the back leg, high rump, underdeveloped haunches and the list continues.
If he is not hard cored program to pace then its either rider/train problems or conformational conditional problems.
One thing I will recommend is that you pick a gait and stick with it. You want him to rack then teach him to rack. Since he sounds like he is performing the stepping pace (aka broken pace) at times then his is going to be easier to train into a rack. Slowing him down is key. I train using a full cheek snaffle and various mouth peices depending on the horse's desire. Perfect the flat walk....use cavelleti, poles on the ground, hills and etc. Something that will get him to begin working his rear and reaching. Thus chainingg hoof flight patterns and conditioning the back. I long line over poles, speeding him/her up and then slowing them down but never breaking into a pace/trot or what have you. Walk walk walk, half halt, half halt etc. Progressing with speed little by little over time. Building up the rear and changing the pulling into the pushing. (ie pulling from the front into the pushing from the rear. Pacey horses will pull from the front and racking/ RW horses will push from the rear more though I have seen shotgun trained rackers pull from the front with thier heads up high and hollowed backs. Not pretty at all.) I only put a leveraged bit in the horse's mouth when they are getting more square and are ready for such.
Feet: yes having a well shod/trimmed horse plays a roll but it should not be a hsort cut to get a gait. I would get a Farrier that is experienced in trimming/shoeing gaited horses. Sometimes the tiniest change can make the largest difference. I have trained horses to rack and run walk with out shoes whith out weights and with out gimmicks.
I recommend an excellent book called "Training the Gaited Horse from the Trail to the Rail" by Gary Lane I love this book and his insights are so SANE. Look it up on Ebay
Another book I recommend (though not as great as the above mentioined) is "Easy-Gaited Horses" by Lee Ziegler (Story publishing)
With out seeing how you are ridding your horse, watching him go and how he is conformed and what not, it is hard to me to give you concise advice. What works for one animal may not for another esp when it comes to bits and tack etc.