I acknowledge that I might not fully understand what an offensive co-odinator does but I think he makes sure all parts of the offense is on the same page running an offense that he has ultimate responsibility for.

I also think he is the designer of plays along with input from the position coaches.

He is the guy who together with the HC decides the game plan each week. Heavy run, heavy pass, what kind of plays (aggressive/conservative), the mix of pass/run.

His play design emphasizes zone or gap blocking.

And finally he is the guy who calls the plays. I see a football game as a chess match. You need to be in a position to set up future plays. I don't see how if you have co's they can both call plays.