Well the easiest way to see it is to go to the extreme again, particularly at the low end where standard deviation is so small as to be easy to ignore. Suppose you depress the run scoring all the way down to, say, .01 runs/game. The probability of anybody having scored through 6 innings is negligible and even, so no advantage there and very little deviation. Basically the odds of the road team leading after 6 innings are near 0 (about 0.3%). The odds of them scoring in the top of the 7th are about 5E-4, so the overall odds of the home team being behind when coming to bat are under 0.4%.
In a high run-scoring environment, there is a significant chance that either team could be losing going into the 7th, and a significant chance of the road team scoring in the top of the 7th. The exact chances are heavily dependent on the standard deviation, but I don't think I need to make up high-scoring math to make it obvious that the odds are significantly greater than 0.4%.