Yes, he did and did an excellent job in the part. I also consider it his best film and other than "Red River" don't remember him playing a character similar to that but he was much more accomplished in "The Searchers."
Neither do I. John Wayne's role as Thomas Dunson in Red River, however, was not nearly as dark as his Ethan Edwards role in The Searchers. Dunson was a driven man, not prone to taking the advice of others, or allowing others their due.
Ethan Edwards, however, definitely had a dark side, something which the movie failed to develop. Some things were implied: Where his money came from, his earlier relationship with his brother's wife, his hatred for Indians. Things such as these made The Searchers a much darker film, and Ethan Edwards a potentially malevolent character.
The look Wayne gives to those two rescued White girls at the military post is close to hatred. It really set up the question of what Edwards will do if he finds his niece. A very chilling moment.