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Find some 4th down from the 1 yard line stats and show me I'm wrong.
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http://www.footballcommentary.com/goallinegambles.htm
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When teams elect to go for it on 4th-and-goal, with the ball officially at the 1-yard line, the observed success rate is around 65%. Since that figure is important for our analysis, we will begin with some issues that affect its interpretation.
In the official NFL play-by-play records (PBP), the spot of the ball is always recorded as an integer number of yards from a goal line. According to the NFL's Guide For Statisticians1, if the ball lies entirely between two yard stripes, the rounding is in the direction of the defender's goal. This implies that whenever the nose of the ball is less than about 5 feet from the defender's goal, the PBP will say that the ball was at the 1-yard line. (The only exceptions derive from the requirement that the spot of the ball and the necessary line for a first down be distinct positive integers. So, if the ball is two inches from the defender's goal with one inch to go for a first down, the PBP will say that the ball was at the 2-yard line with 1 yard to go for a first down.)
Because of the rule for rounding, balls that are officially at the defender's 1-yard line will, on average, be slightly less than a yard from the goal. Offenses do not always go for the touchdown when the ball is officially at the 1-yard line, but even if they did, the observed success rate would be an upward-biased estimate of the probability of success when the ball is really one yard from the goal. However, this bias due to rounding is small.
The more important bias is the "selection bias." The likelihood that a team will go for the touchdown depends systematically on the actual position of the ball. For example, teams are more likely to go for it when the ball is an inch from the goal than when the ball is nearly five feet from the goal. Consequently, among cases in which the ball was officially at the 1-yard line and the offense chose to go for it, we expect that the average distance from the goal was significantly less than a yard.
A similar bias presumably affects the comparison between running and passing when it is 4th-and-goal and the ball is officially at the 1-yard line. In that situation, as Aaron Schatz of Football Outsidershas noted, the success rate is higher for running than for passing. However, one would assume that teams are more likely to run when the actual spot of the ball is closer to the goal. This would explain at least part of the disparity in the success rates. Notice, though, that there is no way to verify this or the previous hypothesis without knowing the true position of the ball.
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Do you have any stats that say that the better team or the home team or the team with "momentum" wins in OT anywhere near 65% of the time? I find that hard to believe.