Being blind or having the eye removed, either way, it is six in one hand, half a dozen in the other, they equate equally. If they are permitted in shows, which I absolutely think should be permitted if it was due to injury, then kudos to the registries. The biggest downfall to a horse with one eye is their owners pitying them and placing limitations on them, I know, I own riding mare with one eye.
It is a horse by horse basis. There are fully sighted horses that scatter every which direction with a slight breeze, and there are horses with limited sight that are physco and others that are amazingly calm. While I was devistated when my gypsy vanner mare lost her eye, so much so, it took me months to even mention it on this forum, she is amazing. The trainer was awesome, the horse is amazingly calm, one of the best riding horses I have owned. My biggest mistake was pittying her and getting into the habit of allowing her to walk all over me. Once I got over that road block, it was smooth sailing. So, to answer your question, the only activities I would be cautious of, at least until you are very familiar with the horse, is driving and jumping.Only because of the added danger of the injury with the cart parts, and depth perception with the jumping, but then again there are one eyed jumpers out there too. Otherwise, the limits are endless.