ASHA (American Saddlebred Horse Assoication) has made it mandatory to have ALL animals registered to be DNA tested and DNA on file. Prior to DNA they used bloodtyping to link foals to parents. I seem to recall it's $50 per hair kit to have this done... so not overly expensive.
I tend to think the JC's rule of live cover only is solely to do with money. It's really easy, and reasonably inexpensive to do a DNA test now... so that would stop most people from trying to "fake" a breeding.
I must say I've found it odd that they wouldn't just "cap" the number of doses a stallion can produce per year (via number of registered offspring by him per year) and allow AI... not only does it allow a stallion to service mares from a lot further away... but it's MUCH safer for him too. (Considering how much a top TB stallion is worth, you'd think his safety would be a priority... that's just me though)