If I extrapolate from my own “noodliness” onto the horse who has a noodle neck, I would say that the horse needs to strengthen the neck muscles. From my own vantage point of having flexibility one area in my body, I would also say that that area is prone to injury, so what I really need is strengthening in that area. Now I know that horses aren’t people, but I do think that for all creatures what is needed is a balance of strength and flexibility.
When I was taking lessons from an amazing cowboy, we’d play around with contact. Granted, his idea of contact was very different from the dressage idea of contact, but the exercises were useful. As someone upthread mentioned, we worked a lot on riding from the seat and legs and not much with the reins. The lessons were a lot about steering that way and defining the shape of the circle with just the legs. If the horse got off course, then maybe take up just enough contact on the rein to show him the way back, but as soon as he got it, then back to just legs again. It was hard work mentally. I remember asking him about this lesson being enough “work” for the horse, because we did most of our work at the walk. His comment was that this was a lot of work for the horse because he had to carry himself and we just had to guide. Carrying himself (with little contact but marching forward, not lollygagging around) was harder for him than letting the rider carry him (a heavier “supporting” contact which you’re more likely to see in dressage), and that built up muscle.
I could go on about his exercises but it would take me lots of time to write up. In any event, I agree that it is about more leg and seat riding and little rein usage in terms of carrying the horse, BUT the horse needs to also be educated about what contact means.
I think that when horses are very talented that it seems easy to go farther with them faster than an untalented horse who struggles with the work. In the situation of the talented horse, one may skip things that seem remedial because the horse seems mentally and physically able to do more work and seems like he “gets it.” Forward and impulsion are great, the horse’s natural outline is great, the horse’s gaits are wonderful, he picks up the work easily and doesn’t seem to struggle with what is being asked, etc. The idea of “contact” isn’t something a horse is born knowing; the lovely horses with great conformation whose necks seem to set them up for easy success can fool us into thinking they “get it” with regard to contact, but really we have to train our idea of contact in them just like anything else.
How does the horse do with in-hand work? Flexions in-hand would be my focus. (Personally I would not put side reins on and lunge a horse like this, but I would do in-hand work before every ride, really focusing on the accepting of contact.) How does he respond to a request for lateral flexion? How does the handler respond? What is the goal of what you are looking for? For a horse that is prone to tension, the handler would look for softness and a release in the neck. For a horse that is prone to noodling around, and is “too soft”, then the handler would look for and reward the appropriate contact. I’d do this standing still, then I’d do it in a circle at the walk (unmounted). Then I’d do it standing still while mounted, and I’d do it intermittently at the walk while mounted, going from walking on a loose rein, to asking for appropriate contact for a couple steps, then back to loose rein, etc. This is what I mean by remedial work.
Some of this might apply to your situation or none of it may apply, but if it were me, I’d go back to some remedial in-hand work.