The ace thing is tough given that Pedro was your classic ace (though declining) but had a bad postseason in 2004. Lester was a very good pitcher up until 2010, but his 2013 and 2012 weren't very good at all (which factors in a ton to the preseason negotiations which fans are now criticizing). Beckett was good across the board in 2007, but his stint in Boston culminated in a 110 ERA+. They were helped out a ton in 2004 by Derek Lowe, who wasn't very good at all in 2003 and 2004. In 2007, it didn't hurt that an aforementioned ace was absolutely horrific in the 2007 ALCS. In the 2013 ALCS, they went against three pitchers who pitched like aces in the regular season, and the ALCS, but they came out ahead led by two starters with a 109 and 116 ERA+.
Out of the Red Sox World Series runs, I'd only call Curt Schilling in 2004 an ace, though Beckett pitched like it in 2007, and Schilling was quite alright that postseason.
While hindsight is great in explaining that teams won with aces, it looks like the last Cy Young winner (which I know is a subjective measure, perhaps a bit less so than defining an "ace") to win the World Series was Guidry in 1978; the last pitcher WAR leader to win was Beckett in 2007 (and before that Randy Johnson in 2001); over the past decade there's not a real pattern of teams winning behind aces only, and going back to 2001 the Diamondbacks don't need 7 games with any bullpen depth at all.
Just looking at last year's team, having a well-rounded club that has a solid bullpen, and a decent rotation can be good enough. Of course they need good luck; what Red Sox squad since 1918 hasn't won without it, and what team since, perhaps, the 1998 Yankees has won without it? I would rather they have some pitchers who are subjectively known as aces, but if they are going to sacrifice depth chasing "aces", I am not sure if it is necessarily the right path to take. It's certainly debatable that the pitchers they have in the organization at this juncture aren't enough, but the "ace" thing is a bit of a red herring. You only have to go back to the 2013 Red Sox, or the 2012 Giants, or 2011 Cardinals, and so on, to see that it's possible to win without a pitcher putting up a season even resembling an ace.