The short answer: no.
Yes, two of the three categories — runs batted in and batting average — are flawed statistics, but that does not erase the accomplishments of a player who accomplishes the feat. After all, home runs and batting average are very different measurements.
The evidence comes courtesy of Andy at the Baseball Reference blog. Using rate stat Wins Above Replacement (WAR), he found that nine of eleven Triple Crown winners also scored highest in WAR. Chuck Klein of the Phillies in 1933 finished fourth and Nap Lajoie of the A’s in 1901 finished second, although the latter was No. 1 among position players.
Some will argue that the Triple Crown findings render advanced statistics unnecessary, but the Klein example refutes such a claim.
Remember, WAR takes into account considerations such as league offense and park factors. Sure enough, one reader noted that the Phillies outfielder played in the notoriously offense-friendly Baker Bowl. Klein’s home/road splits in 1933 produced a ridiculous .467/.516/.789 line at home, while on the road he had a very pedestrian .280/.338/.436 spread.