I have to disagree with your contention that Edward IV was the king in the term ‘kingmaker’. In 1461, it was Edward who had the military victories, the charisma and the right blood in his veins, without which there never would have been a king named Edward IV. Warwick, on the other hand, had the experience and the political acumen to be able to stage the ascent of his young cousin. They were a team and unbeatable. The Lancastrians had nothing to offer against them.
Then after Warwick’s rebellion, as you say, Edward was driven into exile. Warwick pulled Henry VI out of captivity, dusted him off and presented him to the people as their king – again. In my opinion that’s what made him the ‘kingmaker’. He may have been instrumental in putting Edward on the throne and keeping him there in the first years, but it must be admitted that Edward was very active in his own interests. Whereas Henry, in 1470, did nothing to help himself. But for his thrusting Henry back onto the throne he never would have earned that appellation.
Incidentally, it occurred to me that during his first rebellion, Warwick was probably the only man in history to hold two kings captive at the same time. He could as easily be called ‘kingbreaker’.