Every horse has a "gait." The trot is a "gait," as is the canter, gallop, walk, rack, etc.
Every horse's intermediate gait (the one we are talking about when talk about "gait") will fall somewhere on the continnum of gait, as discussed above.
So while it's absolutely true that "Gaited horses aren't trained to gait" it's equally true that a huge number of gaited horse owners train their horses to a gait that may or may not be fully "native" to their horse. If you wanted to do another "continuum" you would put the Big Lick Walker on one end. The other end is tougher because the rider can so easily influence how a horse moves by use of their seat; use of the natural aids (hand and leg); artificial aids (boots, chains, etc.); tack choice and position; husbandry (trimming practices); etc.
Then you've got to put the horse in question on the continuum.
Lots of "breed advocates" claim that their
breed is "all natural." To test their assertion go to a high level show and watch how they move, are ridden, tacked up, etc. In most instances you have to pretty badly streatch the word "natural" to cover what you see.
And even if it is "natural" it still might be harmful over the short or long term. So just hanging the moniker "natural" on the gait does not always tell us what we need to know.
As a rule there's no problem with "polishing" a gait or moving the horse a short distance on the continuum. Just about any combination of tools can be used to achieve this goal. But the rider has to know what they are about when they do this and use some judgement in the process. Whatever program is used should not cause the horse to become sore or lame, in either the short or long term.
Personally, I react pretty negatively as soon as sombody puts the adjective "natural" in front of some aspect the equine art. The word has become so perverted as to not just mean nothing, but to mean the application of some philosophy
that frequently carries with it a lot of "baggage."