This "works flow naturally from faith" idea is only necessary for someone who believes in "faith alone" ... but then you read the Bible, and you find that the square peg doesn't fit in the round hole.
You have to do mental gymnastics, and maybe throw in a little mental tarot card reading, to continue holding to your stance in the face of James 2 ("Abraham was justified by works"), or Ro 14:5,23 (Christians who "sin" aren't "justified" but "condemned"), or 1 Co 9:27-1 Co 10 (salvation and the promise can be forfeited through sin).
The Corinthians, also, were living sinfully, but no one doubts they had "true" faith.
James 4 calls his audience "adulteresses" against "God"--on your view, this should not be possible!
James isn't talking about "true" and "false" faith--when comparing faith to a body, he never mentions a "mannequin", only an incomplete body (without its spirit--dead) and a complete one (with its spirit--living), corresponding to an incomplete faith (without works--dead) and a complete faith (with works--living).
Demons believing doesn't refer to a "false" faith--Jesus never promised to save demons by faith, so there is no amount of faith that could save a demon, the point is they also have "faith alone", and they do not have WORKS.
The man who "says he has faith": James is just comparing the persuasive power of his statement against the persuasive power of a life of good works, and asking which one actually glorifies God, so as to move the audience to works, because glorifying God is the point of Christianity.