My instructor said "go and find out about the nuchal ligament",
so I've been doing a bit of googling.
This is a summary of my findings presented for comment and/or
correction by any lurkers that may still be listening on this ng.

The horse may be thought of as a kind of bridge, with the front legs acting
as one of the pillars, the rear legs as the other pillar, and the backbone
bridging the two. The thoracic and lumbar vertebrae are held together by a
number of ligaments, the topmost ligament (joining the tips of the spinous
processes of each vertebra to its immediate neighbours) is known as the
supraspinous ligament. The spinous processes of the vertebrae in the middle
of the back are short, and get progressively longer as the vertebrae
approach either end (the croup and the withers), giving the supraspinous
ligament and backbone the appearance (and functionality) of a suspension
bridge (with the backbone "hanging" from the ligament on the spinous
processes). This gives the horse's back its characteristic "saddle" shape,
having a dip in the middle even though the backbone itself is not usually
dipped.

This suspension bridge is extended out from its support pillars at each end,
with a short cantilever at the croup (consisting of some of the sacral
vertebrae, and the coccygial vertebrae that comprise the dock of the tail)
and a bigger cantilever at the withers (consisting of the head and neck).
What, then, supports this massive front-end cantilever? If it had to be held
in place by muscle tension, the muscles of the neck would quickly tire, or
else would have evolved to be much more significant (probably to the point
of becoming a hindrance to survival of the animal, and would thereby have
been eradicated by natural selection). The answer is that the supraspinous
ligament continues on past the withers as the nuchal ligament, which
connects directly across from the skull to the first thoracic vertebra,
having "offshoots" reaching downwards to each of the seven cervical
vertebrae. The supraspinous ligament and the nuchal ligament thereby provide
an upper support system for carrying the weight of the head and neck, like
the steel cables of a real suspension bridge. With this support system in
place, the horse can graze and sleep without having to use its shoulder and
neck muscles to support the head and neck. These muscles come into play
when the horse lifts its head from the grazing or sleeping posture and goes
into "sentinel mode".

Knowledge of this support system is essential if one is to understand the
correct way to gymnasticise the horse. In order to allow the horse to employ
the powerful muscles of the hind quarters, the horse must bring its back
hooves closer to the front hooves, thereby allowing them to carry more of
the weight. This action "rotates" the croup ("hocks under"), putting the
supraspinous ligament into tension, and pulling the withers backwards. The
tension is thereby transmitted to the nuchal ligament, lifting the poll,
from which the horse's head may now hang like a chandelier (nose on the
vertical) without employing muscular contraction at the front.

A very grave mistake is to try to emulate the appearance of this "correct"
work by using the reins to pull in the head. To do so is to force the poll
to "rotate", putting the nuchal ligament into tension, which is then
transmitted via the supraspinous ligament to the croup, pulling the croup
forwards. This has the effect of rotating the croup in the wrong direction,
pushing the back hooves further away from the front hooves, thereby
preventing them from carrying weight. The powerful muscles of the hind
quarters are no longer available to the horse, and all gymnastic value is
lost. Furthermore, this outward rotation of the croup causes the backbone to
sag, whereupon the long muscles of the back must carry the weight of the
rider instead of the supraspinous ligament being allowed to do its job (the
steel cable of the suspension bridge no longer supports the bridge). Worse
still, if the horse is asked to perform gymnastic movements in this
grotesque caricature of an "outline", the horse will be in severe discomfort
and may even sustain long-term damage to the nuchal ligament (particularly
where it joins the skull). The horse will find ways to alleviate its
discomfort by tensing certain muscles, and will learn to tighten up whenever
it is taken into the school.

It should be clear from this account why draw-reins are such a controversial
instrument.