Pulling the neck side to side, does cause the horse discomfort.....LET ME DO IT TO YOU!!!!!!!
Um... Speaking as one who has done a lot of CA-type work with my horses in the past, I've got to say that (unless things have changed radically - I haven't seen CA's glorified info-mercial in a while) the lateral flexion exercise isn't "pulling the neck side to side." When I did it, it was take slack out of rein, hold, drop rein when the horse puts slack back into it. No pulling involved. Simply contact, hold, release. No pain, no real discomfort, any more so than applying leg for a leg yield.
Now, whether that exercise actually accomplishes anything constructive is another issue entirely. I've all but dropped it from my own personal routine. I did a little bit with Scout when I first got him, mainly to teach the ORS at the very beginning. If he hasn't been ridden in a while, I'll flex him once each way one time before we move off. Other than that, he doesn't flex CA style. His "flexion exercises" use his entire body. Isolating and manipulating the head alone just isn't going to biomechanically help the horse or prepare him for proper collection, as CA endlessly claims.
True lateral flexion uses the entire length of the body, employed in circles, serpentines, etc. THERE is the "key to vertical flexion" - not that overbent, false give BTV stuff I see so in so many NH trained horses (not just CA, but he's definitely in on it).