Basically, your athletic skill only matters to get you into the pro game. Once you're there, the statistical settings of the playing surfaces makes it essentially a random competition. This is less true in college sports where talent distributions are more skewed, but still plays a role.
Perhaps, but the data you present don't constitute evidence for this proposition. In fact, the only reason they appear to support the above statement is because of cognitive anchoring to talking about probability and especially the probability 1/2, which somehow feels more random then other probabilities. If anything, having the pitcher's distance set so that on average 1/2 the runners make it to first base makes tiny differences in skill more significant.
Agreed, this conclusion is a non sequitur. Regardless of what the average success rate is, some players and teams are more skilled than others and will consistently succeed at a higher rate. People are much better than 50-50 at predicting which team will win a game (and there is a large industry centered in Las Vegas which depends on accurate estimates of these probabilities), which shows that it's not just random.
http://blogs.scientificamerican.com/guest-blog/2011/09/22/cognitive-biases-in-sports-the-irrationality-of-coaches-commentators-and-fans/