Correct. Neutral shoes that have some structure to them, the material will be a little bit stiffer ( especially around the instep and sides of the shoe) to help keep the foot fixed in the shoe, so there is no play. There are some neutral shoes that are overly flexible, like most of the Nike RN Free shoes, where the foot is moving and bending all over the place. It could lead to blisters, sprains, and unstable ground contact. A shoe like that can be a nightmare for a supinator, as well as shoes with weak heel counters. A little bit of stiffness in the heel counter, as well as a little bit of stiffness in the middle part of the shoe will keep that foot locked down.
Shoes with some structure to them, but still have a lot of flexibility :
Zantes
Kinvara
Adios boost
Brooks Pure Flows
zoom streak 6, zoom elite 9, pegasus
Bostons
Adidas DPRs, etc
Shoes that are overly flexible with no structure and weak heel counters:
- The entire Nike Free Line up
- Saucony Freedom ISO - horrible shoe from a structure point. On landing, the foot just bends to the side. It felt insanely unstable. The problem is due to so much weight coming from the rubber sole. If a shoe is really heavy or carries a lot of weight to it, then the upper will need some structure to help keep the foot in place as the shoe slams down due to the weight of the sole. You have this heavy bottom with a very light upper. It does not match. Light shoes have light uppers, and heavy premium trainers have uppers that are more structured. That shoe is totally backwards.
For instance, take a racing flat or a track spike. You would think by the upper being so light and thin that the shoe would be unstable, but not so. Since the shoes are low to the ground, with very little cushioning, the foot is able to have more direct contact with the ground, and the natural pronation of the foot keeps the foot in place. So basically the more cushioning that is added to the shoe, and the higher the shoe is in stack, the more structure is required to keep that foot from moving all over the place. That is why premium trainers have the most structure. They have the most cushioning and the most stack height.
As far as the heel pushing over, if the heel counter is weak or thin, the heel will slide side to side on landing. Think about what would happen if you were to run with your shoes untied, the shoe would be loose, and your landing would be inconsistent because the looseness of the shoe will cause your foot to land a different way on each contact with the ground. That is what happens to the heel with a weak heel counter, and the runner might land more on the outer part of the heel or the inner part of the heel during a run. You will know if that is happening because you will feel dull to sharp pain in those areas while running.