Don't worry about the math behind this, it checks out. As you can see, Braun has no chance. The issue is Posey has so many things going for him that have nothing to do with real baseball things, it's almost stupid. In fact, it is stupid. We cannot measure how much Posey is "carrying the team on his back" or "came through when his team needed him". He's been great in the second half, but assigning other narratives that we can't actually prove with actual numbers or evidence is a faulty basis for determining value. And yes, Posey came back from a horrible injury. That has nothing to do with his current value, it's just a nice story.
Let's look at their numbers side-by-side.
You'll notice Posey is a pretty clear third here, and I'm not just talking about WAR (you'll notice it's called the MVP, not "Best WAR" award). Braun and McCutchen have a clear advantage in hits, home runs, runs, stolen bases, isolated power, slugging, weighted on-base, weighted runs created, baserunning, and yes, WAR. Posey has the edge in.....batting average. Which is the category with the greatest amount of randomness to it. And while runs batted in is a poor stat for evaluating a single player, voters still like it, and Posey is also losing that category to Braun.
"But he's a catcher!", you'll say. Why, yes, he is. He's also played 25 games at first base, which helps him play most every day, but slightly minimizes the value of his offense when he plays there. While WAR isn't the be-all, end-all of player evaluation, Posey is still third behind Braun and McCutchen. And you can't make the argument for Posey due to his position while also making the case for Cabrera (lousy defensive 3B) over Trout (elite CF) in the AL. You either value position or you don't. As a catcher, Posey also has played in less games than any of the other candidates, and while it's a small difference, it still matters. We knock relievers out of Cy Young discussions due to pitching less than starters, and that should also apply here, even if it's ever so slightly a smaller difference.
So what it comes down to is whether or not you think Posey's production as a catcher outweighs the significant gap in the offensive production of the other candidates, Braun and McCutchen in particular. I don't think it does. As far as defense goes, all four guys range from solid to very good, depending on whether you believe the lunacy that is McCutchen's fielding metrics (said metrics seem to have serious trouble with PNC Park). So Posey has no real edge there, either.
Does Posey deserve to be in the conversation? Absolutely, but he's not in the class of Braun or McCutchen (and I'd say it's a coin flip for those two as to who's better), and I might even put him behind Wright. But because he plays for a winner and has no allegations in his past, I have no doubt he'll win it.
In the eyes of this baseball fan, he doesn't quite deserve it.