If you’re just doing a handful of lessons a day, I can understand why you don’t want to make a big investment, but even from a safety perspective, if you’re teaching beginners, I think it’s a good idea to have a fence. What if the kid can’t control a horse properly? Also, not having any access to water an arena will make it hard to keep the footing serviceable over time, during dry and windy spells. Having a wind barrier will also keep the footing in better shape.
I’d personally vote for a nice fence, since the downside with trees and bushes is that they too have to be maintained. Obviously, a fence might lose a rail, but cutting and pruning IMHO is a bigger PIA.
I’ve had a few neighbors who I thought were going to be problems when they initially moved in when they had time on their hands–i.e., constantly knocking on the door, walking through my yard, talking on cellphones, complaining I didn’t trim my bushes frequently enough, even asking me why I cut my own lawn (weekly) rather than get a landscaper. But once they went back to work after getting settled in they just got too busy doing other stuff. It sounds like your guy has way too much time on his hands.
Get a fence, look into getting even a standard sprinkler over to the fence to water things, and then talk over the guy’s “spectating.” I would even put it along the lines of, “my students can be self-conscious, many are beginners, you know how kids are at that age, ha-ha, so shy, hate to have grownups looking at them.”
ETA: I just saw you got the privacy barrier. I hope that contains things!