My feelings on this issue are shaped by the fact that I drive miniature horses which are too small to be ridden and trained any other way. If I had a full-sized horse I might be willing to do only the basic "cantering in harness isn't scary, you don't need to bolt" training then go back to trotting most of the time as they not only cover a lot of ground at a trot but usually have power to spare and can do canter work under saddle to round out their physical training. For my minis, none of those factors are the case.
I do a lot of cross-country and beach driving, where sometimes there are large hills or very deep sand that it's extremely difficult for the horse to pull through at a walk or trot. At a canter he's able to get up enough momentum to make it across these obstacles without undo physical strain. Instead of having to move his load one shoulder at a time as in a trot, he can throw his whole body into the harness and bound forward using the power of his abdominal muscles and coiled hind end against the load. Studies with race horses have also shown that horses breathe better at a canter than a trot as they literally use their entire trunk for inhaling and expelling air as they canter, so my horse breathes better, pulls better, and risks less injury (IMO) by being allowed to canter/gallop in these situations. (You'll notice a tired trotting horse always wants to break to a canter- this is because it's easier for him!)
I also believe strongly in cross-training. Canter work is very good for encouraging a horse to coil their loins and you may have observed that a horse will often offer a very engaged, powerful trot when they first come down from a canter transition. This is a good way to show a horse what he's capable of and encourage a strong trot with proper hind end engagement. It strengthens different muscles, improves their wind, teaches them to move round when done properly, and broadens their minds just as any sort of additional training will. Full-sized horses can get this sort of work under saddle but my minis cannot, so I do it in the cart. I want them to have that sort of well-rounded physical training so their muscles and tendons are both tough and flexible and they are ready for anything.
Just letting them run forward until they break roughly to a canter with their nose in the air and then slop along all strung out teaches them nothing but bad habits, but that's also true of trot work and similar ridden work. Whether under saddle or in the cart you must work your horse properly to get any benefit from it. Canter to me is just another gear and is approached with every bit as much discipline as whoa, walk or trot. My horses are not allowed to break to canter without my permission and as their training advances I get pretty strict about the difference between an extended trot and a canter. They KNOW which one I'm asking for and get in big trouble if they break! They also get in trouble if they stay trotting when I asked for a canter! LOL. I'll actually start schooling specific leads and doing canter figure eights when the horse has some age and experience on them. It gives us one more dimension to practice and push ourselves on.
Besides all that, cantering a mini down the beach is FUN and I enjoy flying around a marathon or cones course switching leads at every bend. It brings back my riding days and builds the bond I have with my horse. Kody loves to run! There's not much as thrilling with a mini as letting him gallop down a windy trail through the woods like his tail's on fire, slaloming through the trees and whooping out loud with glee.
You do need to have the proper equipment though, as many carts are not meant for speed work and would be inclined to tip over sideways if you got going too fast on a turn or didn't counter-balance it properly. EE's in particular are bad for that. Just about any cart can be safely cantered on a straight-away however.
Leia