Work in walk (circles/lateral work) per se does not destroy the walk, and the allows a rider time to think more slowly. It IS easier for the rider too employ aids/time them at the walk. And there is a difference between teaching the rider and training the horse. Many think collecting the walk is just shortening it rather than folding the hind legs correctly through half halts; so they hold/horse slows rather than hind legs folding/they drive to a quicker pace/voila lateral. It IS the last gait to be collected, it IS collected through lateral work (each exercise gathers the horse in a different way--inside hind/outside hind/straightening).
I teach, and do, a LOT of work in walk esp with lateral work: si/t/r on circles as well as straight ahead; not only does it not make the walk lateral, it like slows the tempo.
Walk IS a lateral progression (i.e. Right hind/right fore/lh/lf), it becomes lateral when the forehand is restrained and the tempo is hustled/too quick. It is not necessarily easier for the horse; the problems with tempo/constraint are likely still going to be there as well....less problematic to the purity of the gait? Not necessarily.
So IF a horse is lateral, how to repair it? The most used correction is to put th horse to shoulder in. In most horses gaining control of the diagonal hh will slow it. IF the does not work, then the rider must go to counted walk; sometimes one STEP at a time. But it can be repaired. THE most problematic cause is the fact that riders hold against the horse and create a compressed/low outline. It is also the reason that first comes exercises like a turn on the haunches comes before pirouettes.
Big walks are a good thing in young horses are a good thing allow for relaxation and swinging. Riders MUST (must) allow the horse to bascule and yet be up and VERY open. Flexion longitudinally comes over time. Lots of work in free walk on loose and then long reins, and maintaining the bascules upon going back to working walk/medium walk. IF the horse starts to 'nod' when there is working/medium/collected walk the bascule has been blocked and the half halts have not appropriately folded the hindlegs/kept the arching out to the hand.
I have many horses brought to me to repair the pacing, and for most the 'fix' is fairly easy, then it is about teaching how to sustain the purity to the rider. THE worst case I saw created was by a big name who was going to 'make' a tense horse stand still with a stiff rider, it was ugly. I got the horse to repair; and s.i. Was ineffective. It took counted walk (takes 15 minutes to tour the arena once with one step/hh/another step/hh---and you have a long time to mentally hate the teacher who caused the problem believe me). The stiff rider could sustain the quality of walk then over time and with guidance.
Learning to do a movement IS difficult, it takes a lot of experimentation with the aids (seat or leg before hand or hand before leg or etc). And training a horse means the rider MUST have clear placement and TIMING of the aids. So, it is sort of green on green. But walk gives you time to think...just remember the horse needs to bascule (telescope the head/neck in time with the walk). As long as that is allowed the horse will not pace.