That same argument could be applied to the fourth crusade. If second or third sons of European nobles think that theirs a chance to get their own land and rule their own countries, their going to take it.
The Fourth Crusade was formed around the idea of attacking Egypt. And it winding up in Constantinople was a result of things quite a few of the crusaders objected to.
Plus, we're not looking at second or third sons - at least not on the leadership (common crusader I'm not sure).
So if you advertized it as "Anatolia!", I think the response would be a bit tepid.