There is no reason why a gaited horse of any breed cannot trot, unless he is so extremely laterally gaited he can only pace. I've only seen this in one horse...he couldn't trot at liberty nor even over caveltti in a round pen. A horse that cannot trot is not a flexible horse and is very stiff and hard to ride over time. I have asked a few of the top gaited endurance riders about any horses they know that have completed 50 milers complexity in gait consistently. I am doing research on this as the whole gaited horse in endurance is a pet project of mine.
Gaited horses can cover more ground at their natural walk than a nongaited horse, but they lose steam especially on hills. They are not as efficient as a horse that can trot uphill. Downhill they can make up time provided they don't get pacey. A pacey horse downhill is extremely dangerous..it can fall down way too easily.
I LOVE gaited horses...they are fun to ride, and mine know how to trot as well as gait. BUT I would not even begin to consider a horse with no trot as an endurance prospect. For 25 milers they are fine though.
Training a gaited horse to trot will not "ruin" the gait if you know how to teach a cue to ask for trot or gait. It is not that hard to do...and the horses certainly know the difference of what they are doing with their feet. One of my horses was very pacey when I got her...she could even hard pace. It took almost a year to teach her to trot on cue. Now she trots on cue, or gaits. She's still learning to trot downhill instead of pace but has turned around completely. Her overall topline has improved as she has become more flexible.
Icelandic horses are very gaited and can do a pretty fast speed rack as well as a trot and flying pace. They are required to show all gaits in the show ring.