Fangraphs and Baseball Reference each have their own “implementation” of that WAR framework (which I call for shorthand fWAR and rWAR, respectively). Even Bill James' Win Shares can be considered an implementation of WAR (it shares many of the characteristics).
In any case, with Win Shares, Bill FORCES the individual player Win Shares and Loss Shares so that it exactly matches to a team's Won/Loss record (times 3 anyway). If you were to redo your exercise, that's what you'd find.
With rWAR, the original creator (Sean Smith) and subsequently continued with Sean Forman, they force it so that it matches a team Runs Scored and Runs Allowed (times some Runs per Win multiplier). So, if you were to list the team RS and RA, or even better, the pythagorean W/L record that represents, you'd almost surely (rounding errors aside) see that the sum of the players matches the team Pythag's record.
With fWAR, it's a bit more complicated, but essentially on the hitter side, fWAR would match a team's OBP and SLG numbers. So, if you were to translate OBP and SLG into Runs Scored and Runs Allowed, and then convert that into Wins/Losses, you'd again see that the sum of the players matches the team's translated W/L record. (Not as exact in the fWAR case.)