I would argue the quarterback is the most influential player on the field, almost always.
Well, yeah, the QB is the most important player on the team. Excluding the coach, GM, and owner, he may be the most important person in the entire organization.
A bad QB = a bad team, and a bad (minus QB) team and a great QB can = an okay team
Eh, here is where you start to make the leap from "most important" to "only thing of importance". The 2000 Ravens won a Superbowl with Trent Dilfer at QB. Meanwhile, guys like Philip Rivers and Dan Marino, both very very good QB's, have never won a Superbowl.
What's more important than having a good QB is having a system that your QB can thrive in. Tom Brady, if playing on the Bears for the last decade or so, would be far less successful. That's because Tom Brady relies on good pass protection to be successful, and the Pats have been smart and kept a great line in front of him for the vast majority of his career.