It makes me angry how many times i see pro equine photographers being snappy with people who are just out to learn, not to steal business. and anyway, If you're good enough, IMO your business won't be stolen.
Just a couple of things to think about; If you were to find a wedding going on in a public park, would you consider it ok to start moving around in the area of the wedding and the wedding party and guests, and shooting the wedding in order to learn wedding photography?
Can learning take place by contacting a local trainer's barn, and asking for permission to come and practice during lessons and training sessions - or can learning only happen at the shows?
If what you want to accomplish can only be done during an actual show, how about practicing and learning in the warm up rings (with the prior blessing of the OP)?
Side note; when I was learning, I found many OPs who were willing to help me to learn. Even found a couple who offered to let me "test drive" some of the equipment that they had, that was on my "to be aquired" or "wish" list. And I found some who actually had the equipment that I had back then, as their backup gear! Others who, during breaks would take time to review what I had and make helpful comments to help me to improve.
The difference was that I tried to be courteous and contacted them ahead of time, explained what it was that I hoped to accomplish, and asked if it would be ok; if it was, then I introduced myself at the show before I did anything else, confirmed again what it was that I hoped to accomplish. (If it wasn't OK with them for any reason, I refrained and found another venue).
Think about other reasons that an OP might be uncomfortable with someone who just shows up and starts shooting over the rail; one of my horror stories involves a photographer who ultimately unzipped and proceeded to relieve himself - in view of the grandstands and in front of a class of junior riders ! Unfortunately many exhibitors who did not know me, assumed that this man was working for/with the OP - in this case, me!
Just some things to think about.