In general, the European warmblood registries developed over the course of the 20th century to breed very high quality sport horses (show jumping, dressage, eventing) from mixing Thoroughbred blood with the local breeds of cavalry and harness horse. The aim was to get a horse with TB speed and heart, but more height, bone substance and in some cases much flashier gaits.
Breeding these horses was and is supported and regulated by governments as a national industry in a way that doesn’t happen in North America. Many of the registries reflect the German town or region that the foundation stock came from such as Oldenburg, Hanover, Holstein, Westfalen, Trahkener, which were principalities with their own high end cavalry and coach horses before the unification of Germany. There are also WB registries in the Netherlands and Belgium, and of course now in many other countries but using horses from these Northern European breeding programs
The breeding program has been very successful, such that these countries tend to dominate in international equestrian competition and also that warmbloods have become pretty much the required kind of horse for high level jumping and dressage. I understand some riders still prefer TB for eventing because of sheer speed.
The registries remain open so that excellent horses of other registries and also some TB and even AngloArab horses (in the Selle Francais) can be included as it seems useful to develop the breed. I understand too that even if you have a foal from two registered parents, it needs to pass inspection before it can be registered itself.
However, while WB dominate at the highest levels of horse sport, it’s also true there are a lot of mediocre WB, especially in North America where breeding is not regulated. However, even a mediocre WB commands a price premium over say a TB of equal quality, and it is also true that even a mediocre WB probably has more talent for competitive dressage than other breeds, since the rules of dressage have to an extent evolved alongside the horse: the emphasis on big gaits is an example of this. Take two equally well trained horses, and the one with the larger gaits will score higher on the same test, meaning that people will pay a premium for even a mediocre WB (mediocre compared to top WB).
In North America, the term “WB” can get diluted alot. People will call a TB/draft a WB, or a TB/WB when the TB parent is not accepted into any of the European registries. And of course NA has various WB and Sport Horse registries that do not have very strict entry requirements, compared to the European registries.