There have been one-eyed horses among the top performers in racing, jumping, driving, eventing, endurance, and just about every other equine occupation. I would recommend continuing to ask around to get advice or help from people who have started a young, one-sided vision horse to get some firsthand tips.
My current horse has only one eye and you’d never know it unless you look right at him. He’s not spooky on that side, no issues except that when you longe him to the left (right eye is gone) he does tend to come in on the circle a bit. He’s fine to the right because he can see the round pen on that side. Prior to my owning him, he had been a pony club horse, safely jumping up to 3 feet, and he’s confirmed at 3rd level as well. I don’t treat him any differently than any other horse; he doesn’t act any differently. I DO always talk to him when i’m approaching from the blind side, but i don’t have to. I do it to be courteous and not startle him, but he’s never spooked even if i’ve come up on that side silently.
I think as long as you don’t make a huge issue about it and get nervous yourself, he’ll be fine. Most horses adjust really well to only having one eye.