To clarify, I don't think the in game managing is as important as the off field managing. The in game decisions have little impact. The managing of the players and the clubhouse, ie, the human element, , has a bigger impact.
I do not have any proof of this other than anecdotal evidence, but playing for a manager that is respected by and respects the players, and that they know has their backs, goes a long way in terms of on field production. Likewise, playing for a manager that will throw you under a bus has a strong adverse effect on the team's play. Case in point - Bobby Valentine.
I think Farrell does a good job with the off the field stuff.
All that said, I don't think managers should be making millions. Then again, I think players are way overpaid too.