I'd say it's mostly true, but it depends on the horses conformation.
I have a pretty large appaloosa that weighs 1130 pounds, and he can carry a grown man (250 lbs+).
I also have a tiny Connemara pony who weighs about 850 pounds and is probably 12 inches shorter and he could carry just as much as the large horse. In fact, Connemara's are known for being weight-carrying ponies. Donkeys are the same way. However, it depends on their bone density. My Connemara is super dense, stocky and sturdy like a rock. There are some ponies that would be his height that may only be able to carry a small child because they're quite frail, if that makes sense. :)