In answer to the posters question, no they would not do well in an APA sanctioned show.
To answer the others persons question about "Why not breed pure breeds'
In the case of Easter eggers, this was done purposely because people wanted the different color egg possibilities, rather than just blue.
In the case of other breeds. I think if you track back you will find that hatcheries did begin with Pure breeds, however maintaining the proper standard require constant culling , especially when using in line breeding. Why in line breeding need to be used is simple, their wasn't and still isn't enough diversity in pure breed stock. If you trace the origin of the birds back to the original breeders that brought the stock over from Europe you will see you are dealing with a very small starting flock.
Hatcheries stay in business by selling, not culling chicks, so they started cutting in close breeds to increase the genetic pool, this helps maintain egg size , laying patterns and health. Thus the "Industrial or production" breeds began.
Thats why it seems to confusing to some, they go to one of the conservation sites and they see breeds listed as endangered, yet they go to any of the hatchery sites and find out you can order as many as you want.
Hope I answered the other persons question.