Standardbred registries have been allowing AI for years, ever since that huge mess in Kentucky over equine STDs so long ago. One can have semen shipped from any farm that ships, no problems. We have also been doing DNA parent verification for ages and blood typing long before that. Embryo transfer is also allowed, but only one foal per mare per year, no multiple foals from embryo transfer mares. I think the JC is long overdue for an overhaul - stepping into the 1970’s would be a huge step for them; expecting them to step into the 21st century might be expecting too much though - culture shock for the Old Boys Club.
In any case, DNA or no, one CAN get into parent verification problems with live cover as well: this scenario happened a few years ago - mare died when the foal was 3-4 months old and the breeder forgot to save some mane hairs and registering that USTA horse (DNA not required then) to SC was a real headache for the man. All that saved his bacon was he had a half sibling to that horse, and managed that way to prove who the mare was by marker eliminations.