The Cardinals are obviously a very good team, but some of the mistakes they make as an organization would draw much more attention if not for the best organization frame. One of the most memorable moments of the last World Series occurred in game four when Kolten Wong got picked off first base for the last out of the game with the tying run, in the person of Carlos Beltran, at the plate. It is not fair to blame that defeat on Wong as the Cardinals were in a very tough position, but if the top prospect for another organization had made such a mistake in a similar situation, it would likely have been viewed as a reflection on that organization. Similarly, failing to address the shortstop situation after 2012 was an organizational mistake.
Clearly the team that wins the World Series plays the game the right way, but the meaning of this phrase in not always what it might seem. The right way to play baseball at the big league level is to score more runs than the other team. That is all. However, when an announcer or writer describes a team as playing the game the right way, this usually means the team fields well, doesn't hit too many home runs, probably bunts too much and does things like move the runners by hitting to the right side of the infield.