As far as riding is concerned, even basic under saddle work does not vary much between a trotter and a gaited horse. There are more complexities with getting a specific gait than there are getting a generic trot, but when a horse first goes under saddle the wise trainer looks for a quiet, forward, horse yielding to the bit more than any sort of specific gait. If you get that gait, then God Bless You. If you don't, it's not a concern at that time.
I agree and disagree with this at the same time LOL
While I think any trainer should be able to start a horse...after 4 months you should be getting into gaiting.
I understand that *any* trainer should be trying to get basic skills down first before gaiting. My point is, if you send a gaited horse to a regular training to be broke and that horse starts off gaiting incorrectly that trainer may not know how to correct the problem. If a horse gets pacey, it needs to be corrected. The sooner the better.
If you didn't care about the correctness of gait or the smoothness of the gait...you could send your horse to whatever trainer fit your needs. But then again...if you didnt care...why buy a gaited horse at all?