Horse trainer building is said to be unlocked by national horse breeder period.
If i have to move some tamed horses around yet still, title national horse breeder is stupid and misleading. It really should just open horse trainer buildings in all cities just like pedia of horse breeder claims it will. So there is another misleading pedia"info".
This build horse and move it around-bs is just useless and pointless. Horse breeder-titled building could be fathomed to act like this but not national horse breeder. Especially when national mints act as they promise and claim.
Just bad designing with false pedia info, luckily not anything game breaking.
They are functioning the exact same way as the national metal smelters are. It provides a resource for the entire nation. The national Iron Smelter does not unlock Iron Mine in all cities for example.
Your real gripe, the one that you are almost unable to communicate properly to this team; is that there are buildings that, in your opinion, require a resource in city vicinity when it should only require the resource.
Your initial bug report is about the animal trainers requiring a local source of the animal they train; not about the national horse breeder being broken.