Would he though ?
From what we've heard he is very cautious in the transfer market, often to the point of absurdity. He took zero interest in youth development at Everton (a major part of directing a football club, surely), and a large number of Everton's back room staff seemingly thought he was an utter nob.
He likes to have complete control, something which being a director of football wouldn't give him as pitch matters would be out of his hands.
He's suited to managing a mid table Premier League side.
Can't criticise his record in the transfer market at all mate - he's phenomenal, and I'm his biggest critic. His success rate is unmatched. Can call him cautious all we like, but he got sensational results. Indeed, it was his track record in the transfer market which hid all his other major faults that have been ruthlessly exposed at United.
A Director of Football isn't the same job as academy manager or under 18s coach, and it'd be ultimately up to the manager what players are good enough for the first team. As a Director of Football, his job would largely be recruitment, and I can't think of many better in world football than Moyes at that specific job.
You could have a point in terms of his ego, and he'd have to learn to accept what the Director of Football role entails, but if he could accept having solely the responsibility of incomings to a club rather than the utilisation of that talent on the pitch then he could have massive success IMO.