Education back then was built around apprenticeships and practical learning with classroom learning only being available to the super wealthy (which Washington was a part of, so it's likely he was very learned. I have spent a lot of time studying the History of Education and to compare what someone went through in the 18th and 19th Centuries to today is intellectually dishonest. Washington became President because there was no one else that could be. After the war, he was by far the most popular individual person in the country. And seeing as one of the primary roles of the President is Commander in Chief, Washington was a uniquely qualified as he was the leader of the Colonial Army.
No one is saying that you can't be intelligent without college. It sort of feels like you're trying to quell a personal anxiety, but there's no reason to. The most successful person financially in my family is not college educated. However, for better or worse, college is a stepping stone on the path to many other jobs, political office being one of them.