I like the batting average example.
You have two players. One has the following, 285/1000 and the other 4/4. First, a little thinking. The pair's average batting average should be in the neighborhood of the more prolific player. The player who barely gets in the game should have very little influence.
If we average the averages (your first method), we get (285/1000 + 4/4)/2 = 0.643 -- This should not make sense to anyone.
If we examine the totals, both hits and at-bats (your second method), we get (285+4)/(1000+4) = 289/1004 = 0.288 -- Much more reasonable.