I’d bet dollars to donuts you could trace horseback riding recurring throughout the series to DeForest Kelley. DeForest was a well established actor in the Western genre before his role in Star Trek. And at a certain point he was getting screwed on the royalties of the Star Trek films, so William and Leonard pushed to get him a producer credit on the next film. That film also had a camping in wilderness tribute to DeForest at the beginning. I think the first horse scenes were Kirk and then Kirk and Picard.