I really was going to title this A Rallying Cry for Classical Dressage, but I can foresee (and understand) eyes starting to roll and attention to drift immediately. That's for one because virtually every trainer calls themselves classical, even those whose methods would have a hard time qualifying. And secondly, it does conjure the image of skinny men in uniform with 12 years of minimum training on white Lippizaners which is admirable but relates how to my 17.3 warmblood??
So instead, perhaps, just think of it as honest riding. Riding that is not starting with the hands of the rider but with his/her own body. I don't care if you ever want to "show dressage" or even desire to "ride dressage." I'd be happy and so, btw. will your horse, if you ride honestly, ride well, giving him a chance to travel straight through his body by using lateral work and other exercises to address his natural crookedness (that are well detailed in the literature, but you can also find a lot of essential info in Andrew Murphy's blog equinerds.com), swing through his back and stretch forward into a soft contact.
Why is that important, especially if you ride in another discipline? First and foremost it is about the longevity of the horse. Crookedness (natural left or right "handedness" and other movement patterns particular to each horse) leads at the very least to soreness and often also to lameness. This is well documented in veterinary literature. Also a horse who is not lifting his back in his work is prone to develop kissing spines amongst other problems.
But a horse who is allowed to work (toward being) properly aligned in his body also becomes a more relaxed horse, a more willing horse. My job is to specifically address crookedness issues with all my horses and even those who I ride only very infrequently surprise me with remarkable retention of what we've worked on before, even if there have been weeks in between rides. Horses who have a tendency to come out tense in their backs learn to let go quickly.
The two most valuable mantras for my own riding are a) what my trainer Tanya Vik keeps telling me: "The horse can only go as well as the rider can sit." and b) Andrew Murphy's "Don't stabilize anything that's wrong." The latter in particular is tempting to ignore on occasion, because of course I could overbend the horse a bit and make him appear "round" quicker, but at the same time I'd deny him the chance to stretch properly, align properly, and figure it out for himself. And the difference between a pulling back rein and a horse taking the contact forward (ultimately creating self carriage) is like the difference between you memorizing a chapter in a philosophy book and you actually taking the time to understand the content.
This kind of honest riding that's been described in the classical literature builds true partnership with the horse, helps him develop and overcome problems in his body (and occasionally also in his mind). It literally is rehabilitative. And in terms getting this rewarding feeling when you finish a ride, nothing, absolutely nothing feels better than a happy, completely relaxed horse who is at peace with the world.