I would probably try doing lots of walk-trot-walk transitions with only a few steps in each gait. That way he'll start to expect a down transition and may not feel the need to burn rubber. Once (if) he gets this, make the intervals between transition longer and, most importantly, no rein pressure as long as he maintains speed. If he does speed up, down transition to walk - not punitively or as a correction, just as a matter of course. The positive reinforcement comes when he does maintain speed, you stay in neutral, give him an affectionate rub on the neck, and tell him he's the best thing since sliced bread. If you really want to drive the point home, call it quits as soon as he maintains speed on a loose rein for an entire arena length. Then build on that.
Consider the difference in footfalls between the walk and the trot. The walk is a four-beat gait with three feet generally on the ground at any one time. The trot is a two-beat gait with two feet on the ground at any one time and a period of suspension between the beats.
Your horse may find the trot more fun generally. On harder surfaces, however, he may choose the running walk since he can spread his weight over three feet reducing any discomfort from impact.
Gaiting on the flat surface is generally "easy", whereas doing the same in the grass may just require enough of additional strength and coordination throwing him off and he choses the easier gait - trot.
How fit is he?
He is fairly muscular, a little fat in the belly. He has more shoulder than hindquarter muscle and his underneck is very very bulgy- most likely from being ridden in a rack for many years. He was a show horse for his former owners so I KNOW he knows how to gait well.
Some horses prefer to trot. My foxtrotter has a huge trot and that is her preferred gait. She did not foxtrot at all until i had conditioned her for 3 months. Now she will foxtrot but i have to half halt and hold her in the gait. She struggles to hold the gait if conditions get rough- uneven footing, roots, or branches down, or hills, often result in her breaking gait. I try to pick ideal conditions when asking her to gait. She tends to be lazy and gaiting is work.
I like having the option of gaiting or not. I expect her to do both.
I would love to have him both trot and gait on cue someday, but first I want to eliminate the trot so he can do a proper runningwalk
Find the spot the horse gaits, and only do the gait. Make a cue for it....shift your weight back, lift your hands a bit, whatever works for you. Go ONLY between walk and the gait. If he breaks, return to walk, and start over, emphasizing your cues. Turning a baited horse in a circle is counter productive. The gaits are laterally based, and difficult to maintain in a small circle, but a trot, being diagonal, is easy on a circle.
As he gets set in his gait, go from your smooth surface to grass, only long enough to test ar first, then gradually add time.
This is what i'm trying at the moment. Just gaiting and gaiting on a smooth surface and reinforcing when he does, and moving that cue to in the grass. Thank you