I looked at every World Series winner in the Wild Card era (1995-present) and how many batters their "most used" pitcher faced, and then how many plate appearances their "most used" hitter had. There was usually a very wide margin between the two, always in favor of the pitcher.*
The "ace" pitcher faced an average of 136 hitters in the postseason (they ranged from Greg Maddux to Andy Pettitte to Josh Beckett to Chris Carpenter). The most used (usually leadoff) hitter had an average of 70 PA/postseason, just about half as many opportunities to help his team as the pitcher.**
So it's pretty obvious that a single pitcher does, in fact, play a bigger role in the playoffs than a single hitter.
But what if we include a hitter's defense?
Going by regular season stats, there have been 37-38 defensive chances per game over the last 17 years. Dividing that by nine fielders equals about four chances per player per game. (Obviously this number can vary based on position: While catchers, first- and second-basemen have the most chances, pitchers have the fewest; but it's just an average, so bear with me.) If we add four "opportunities" to each hitter through each playoff game, where does that put them?
It makes it much closer. It gives hitters an extra 60 opportunities (on average) to help their team in the postseason. That brings a position player's total to 130, only six shy of pitchers' opportunities. Throw in a few stolen bases and it's nearly even.
But... pitchers also contribute on defense (not a lot), about two chances per game. But they only appear every few days or so, so it wouldn't add all that much to their totals.
In the end, it appears that a single pitcher can (and usually does) contribute more during the playoffs (what with extra off days and less worry about late-season health) than a single position player. That, of course, doesn't mean he will (the ace pitcher could suck), but he does have more opportunities to do so.
Examples: In the 1999 playoffs, Orlando Hernandez faced 117 batters and Derek Jeter had 53 PA. Toss in about four plays/game for Jeter on defense (over 12 games) and he gets credit for 101 opportunities, still less than El Duque though. This past October, however, while Chris Carpenter faced 148 batters over St. Louis' 18 postseason games, Rafael Furcal had 81 PA and approximately 72 chances on defense, for a total of 153, slightly higher than the pitcher. But this was the exception, not the rule.***
Perhaps Brian Cashman was thinking along the same lines when he traded Jesus Montero for Michael Pineda. Since Montero might have been nothing more than a DH with the Yankees (and would therefore receive zero defensive credit), the fact that Cashman traded him for a (potentially great) starting pitcher makes sense.
* All I cared about was who had more opportunities to help his club, not whether he did or not.
** Instead of choosing one hitter, I sometimes chose a combination of hitters that had the most PA in each playoff series (LDS, LCS and WS), so while in some years that was one player, other years it was three. For example, Tim Raines led the Yankees in PA in the '96 LDS, but Derek Jeter had more in the LCS and it was Bernie Williams in the WS. In '98 though, Paul O'Neill led the club in every series. If anything, this gives slightly more credit to the hitters than they deserve.
*** Go back to the days of four-man rotations and the "ace" pitcher gets an even bigger advantage.