Don't know about the U.S.
Primarily I have a Hungarian Sportshorse and have had a KWPN Stallion.
Both are fully papered but neither has been presented to or passed the registration - so therefore are NOT in the studbook. Both however are in the breed register - I can contact the breed associations and trace both horses .
I have had many ' purebred ' horses that were fully papered but not in their relevand studbooks - this does not mean they are not examples of their breed - just that they haven't been graded.
Examples that I have had that were fully papered.
3* hungarian Sportshorses
Welsh Section A
All of these ( except the Shagya and Welsh A ) are considered to be warmbloods