In the OP you said it unlocked after your first competition, and it's resetting everyday. So it would stand to reason that that's due to it showing competitions your sim can compete in and not your horse, since the horse can't compete anymore that day.
I.E. they are showing unlocked because a horse is not selected. Bet if you have a second horse. It would show which ones the new horse qualifies for and lock the rest, once the new horse is selected.
This explains why it's resetting and why it showed that you had no horse on the lot after your first competition.
Also when you first click 'enter competition', before you select one, on the right side of the pop up is where it says horses can only do one competition a day. If you look at the photo the other guy previously shared. It's the 4th sentence on the right side.
