I know that some shows do have a Bred-by competition where you can check a box on your entry and CHs are eligible to compete, but why don't CHs compete for Best bred by at specialties?? I was just wondering if there's a reason for this and it seems odd to me that only class dogs are chosen from.

I thought it was just a competition of those that were competing in the BBE class.

Yes, I beleive it's considered for the regular class winners only.