Obviously lunging in a caveson has advantages: the tender mouth/bars/tongue will not be affected and the horse can be used to working into pressure with less drama and pain. Ideally the handler has a D or full cheek/fulmer for starting horses (prevents unilateral use of the bit/pulling through. Additionally the head is guided slightly inward and vibrations/hh work more clearly. There must be a jaw strap (not a throat latch strap) to prevent it from being pulled into the outside eye)
Using a halter means either the handler is either pulling down (if attached underneath) or inward (if attached to the inside ring). THe later also prevents changing directions easily.
A caveson also assures that the bit cannot be pulled through the mouth (for those that just attach to only the inside ring) or the action being too strong a half halt (the bit IS correctly LIFTED in the mouth when it is run over the top) but it does not easily release.
So, going back to the use of side reins: IF the horse does not stay active in front of the vertical the bit (esp a double broken snaffle) correctly working into a light connection, and if it is closed (to or btv) in its throatlatch or lower will action both on the tongue and the bars (often people use sliding side reins are better because the horse is more closed/lower.
The point of s.r. is NOT longitudinal flexion, it is light lateral flexibility (even through the body). And the horse learns how to take hh (light vibrations on the caveson) with a steady conneciton on the lunge lines. What are the purpose of hh? To get attention/prepare for gait changes/keep the horse up and open and connected. And since NO side reins allow for the bascule/telescoping of the neck in walk or canter, they are primarily for work in trot with very brief periods in the other gaits.