Ooh, interesting thread.
I called the AQHA about registering Charity with them (Charity is a registered paint. Her sire is Reg. APH, and dam is Reg. AQH. Both are pure not appendix or anything).
They said no, because "Paints are a color breed". That is absolutely false. The PINTO horse association is a color breed, as the horses registered can be gaited, trotters, ponies, etc. the ONE thing they all must have in common is the COLOR. If the APH was a color breed, they wouldn't restrict registry to APH crossed with TB and AQH. The body type that the Paint horse is required to have needs to reflect that of a TB or AQH.
MY THOUGHTS.... so hear me out... Way back in time when ranchers were breeding their AQH every so often a baby would be produced with splashy white markings. This was looked upon as a defect in the herd, and reflected poorly on the ranch's quality of horses and the baby is unwanted solely due to color. Someone felt bad for these horses, because they were still GOOD horses, just flashier in color.. so they started the APHA. SO THEN............. fast forward to present days... APH breeders get a solid baby in their herd, and its looked upon as a defect, and it reflects poorly on the breeders quality of horses, and the baby is unwanted solely due to LACK of color.
SO.. my question is...
Isn't a Breeding Stock Paint just a Quarter Horse?! Both registries say No. But, I mean.. the breed standard HAS to be close, right?!! Both the AQHA and APHA use TBs to breed with their stock. And the APH uses ONLY the AQH and TB to make their breed.
My head huuuuuuuuuuuurts ;__;
I think there is a bunch of bad blood between the AQHA and APHA, after talking to both sides, lol. Buncha Grumpy Gus'! ;)
I am the proud owner of two Breeding Stock Paints

Charity and Skippy! <3 and PSST... Charity looks just like a AQH :O Skippy! Has the wonky blaze, so he looks more like a Paint ;)
I think the reason they are called "Breeding Stock Paints" is because they may not have the color, but they have the build/conformation that a Paint horse should have. Therefore, if you breed a BSP to a Colored Paint with poor conformation, the baby will hopefully have color, AND the build of a Paint Horse.
Charity is what I would consider a BSP by my definition. She is Half paint and Half Quarter Horse... So she has the stronger, less diluted build of a AQH, and if bred to a stallion that is awesome in color, but poor in conformation, she may be able to "fix" it in the baby.
I really REALLY hope that makes sense, LOL! I confused myself a few times while writing it ^^;;;