I have 5 or 6 horses being managed at this moment, but none are showing for some reason. ALL have riders who are paid, ALL get trained once a week along with riders, ALL are set up to show for Monday's, Wednesday's, and Friday's, and I have almost 300,000 ebs so the money isn't a problem. They are only being entered in shows when I manually enter them, which defeats the purpose of having the manager. Am I doing something wrong or is there something wrong with the game?

Not a bug. The BM only enters horses that are ready to show if you have her set to competitive only.

A horse doesn't show every week. There are breaks a horse takes when it levels up for their training bars to become high enough to catch up to their new level. This can take several weeks so if you only have 5-6 horses showing it's very possible that they are all on their "break" period waiting for their training bars to increase. I'd suggest at least getting a full barn full for a more constant income :)