It might be helpful to distinguish between "legal rights" and "natural rights".
Legally, it would be tough to prove libel. And I don't think you have a legal right to privacy when you do something in public - unless you are a cop in some states, where you can be arrested for videotaping a cop in his official duties.
But rights can also refer to natural rights. Natural law - common decency and a concern for the rights of the accused - says a man has the right to face his opponents. Posting a video on YouTube calling someone an animal abuser violates that natural right, even if our law doesn't give one legal recourse.
Freedom of speech works best when people speak responsibly. Making a public, anonymous accusation of animal abuse, with a 70 second video to back it up, is very irresponsible.
The Arizona law for animal abuse is at this link: Format Document
A. A person commits cruelty to animals if the person does any of the following:
1. Intentionally, knowingly or recklessly subjects any animal under the person's custody or control to cruel neglect or abandonment.
2. Intentionally, knowingly or recklessly fails to provide medical attention necessary to prevent protracted suffering to any animal under the person's custody or control.
3. Intentionally, knowingly or recklessly inflicts unnecessary physical injury to any animal.
4. Recklessly subjects any animal to cruel mistreatment...
...2. "Cruel mistreatment" means to torture or otherwise inflict unnecessary serious physical injury upon an animal or to kill an animal in a manner that causes protracted suffering to the animal."
That has nothing to do with "I think the guy is too heavy handed with the reins", or even, "I don't think people should use bits" or "I don't think people should ride horses".
When you allow people to define animal abuse based, not on physical harm, but on "what I don't like to see", you open up the door to allowing accusations of abuse for owning a dog, riding a horse, rounding up cattle in the sun, etc.
Arizona also says:
"13-2910.05. Exempt activities
(Caution: 1998 Prop. 105 applies)
Activity involving the possession, training, exhibition or use of an animal in the otherwise lawful pursuits of hunting, ranching, farming, rodeos, shows and security services shall be exempt from the provisions of sections 13-2910.01, 13-2910.02, 13-2910.03 and 13-2910.04."
The law would vary from state to state, but I'm not comfortable with the idea that anyone who wants to accuse you of cruelty to animals (Do your horses ever sweat while riding?) should be able to do so on Youtube, and potentially cost you employment etc at a later date.
This is a case where the law hasn't caught up with the extensive defamation that can occur in hours on the Internet.