Iberian horses, times past, were the Rolls Royce of war horses, exported across the world. They are still used to work cattle, show off in parades and used extensively for leisure riding. They are intelligent, sensitive, kind, spirited, trainable. Naturally, they are close coupled. I’ve ridden some where their four hoof prints are covered by an area about the diameter of a dinner plate! Gallop is an uncommon gait in Iberian horsemanship. The local terrain in Andalusia and southern Portugal is often hard and stoney so natural collection is a better option. This means extended gaits can be harder for Iberians in competitive Dressage. If you watch working equitation, or even better, a Portuguese rider working a bull, it is possible to see most of the dressage movements in use.
The European Warmbloods were bred to be army remounts in times of war and draft horses for farm work or pulling carriages at all other times. Sufficient horses for war was a Government priority for centuries. Quality stallions were kept by the State Stud farms while the farmers kept a few mares to breed and had geldings to work. Sport only became the focus of European WB breeding after WW2. These horses are reasonably intelligent but not necessarily the sharpest knife in the drawer because troopers had to be able to handle them, have a longer body shape designed to pull things, they are trainable and often people focused. How “hot” they are often depends on how much hot blood they have i.e. TB or Arab. Some “modern types” have a high percentage of blood and can gallop. Both France and Germany had large armies, often based on obligatory national service, with cavalry training to a high standard. This has shaped the modern discipline of Dressage. WB tend to find collection harder than extension and competitive Dressage tests are written to suit this type of animal. Iberians tend to start to shine at higher levels where increasing collection is the objective.
The TB, not in OPs question, has been tested for centuries for speed. They are supreme athletes, intelligent, brave and enduring, which is why they have been used to “improve” all the other breeds in Europe. Even in Spain, which treasures ancient bloodlines, uses TB and Arab stallions to obtain a more functional sport horse. The fastest growing section in British Dressage is TB dressage because we now understand how dressage should work and so can apply it to TBs and other “exotic” (German terminology) breeds.