I think it's safe to say that the networks have no influence over the outcome of NFL games. If they did, you never would have seen so many bad ratings-generating teams make it to the Super Bowl in recent years (Carolina, Baltimore, Tampa Bay, etc.).
That's only because NBC hasn't been in the mix the last few years. Now that they're there, expect a heavy dose of teams with "New" in their name, especially in years that NBC broadcasts the Super Bowl.