Again, you must pay attention to both sides of the story. You can never overthink things unless you come up with the wrong conclusion.
I do not necessarily agree with the notion that you have to have an "identity" on offense to be good. Tell me, what do you think was New England's identity on offense in all their super bowl seasons? They ran the ball well, they passed the ball well, and they did either depending on what the defense was allowing them to exploit. IMO, there's nothing wrong with having an offense like that because it means you're good at whatever you do. You could be a passing team, you could be a running team. That's the best way to put a defense off balance and it's why you see the Patriots be able to come back like murder on teams in the 4th quarter.
I agree with what Linehan wants this offense to be. But, at some point, you have to start paying attention to what this offense IS rather than what you want it to be.