There are good reasons why the A's don't do as well as other teams in the playoffs. Mostly, it's that "playoff" baseball is different than regular season baseball. The strike zones expand, the scores drop and the breaks between innings increase to allow for more commercials. There's also more rest/travel days between games. All of this magnifies the value of top pitchers since it allows them to pitch a greater percentage of games, pitch longer in those games and pitch more effectively in those games. Meanwhile, the value of the pitchers at the bottom of the rotation is minimized or eliminated.
One of the reasons the A's have won so much in the regular season is that they make up for the lack of high-priced stars by being solid, top to bottom. As evidenced by this season, when two of their starters went down for the entire year, they had guys in the minors who were almost as good and the drop off was minimal. But whenever one of their guys starts to look really good, they trade him to replenish the minor league stock. It happened with Hudson, Mulder, Haren, Gonzalez and Cahill. This incredible depth of similar players works really well in the regular season and makes the A's pretty much injury-proof. But it works against them in the playoffs when 1 stud pitcher can basically dominate them as we've seen with Justin Verlander.
They've been eliminated twice by him in games where he basically pounded the outside corner on a pitch that isn't a strike but the umpire was giving the entire night, albeit to both pitchers. But since the A's don't have a pitcher that can hit that spot with near-100mph heat reliably, the Tigers were able to take advantage of a few location misses. When the schedule means that you're facing Verlander for 40% of a series and the extended breaks between innings make it easy for Verlander to pitch the entire game, it all adds up to the A's getting beat and, I think, explains why Beane went out and got Samardzija to pair against Verlander or whoever that #1 guy they face is thinking that both of them can pound that outside-the-zone strike with upper 90s stuff equally well.
Also, the OPS-minded strategy doesn't work as well against #1 guys as it does against #4 and #5 guys who you're not going to face much in a playoff series. And with lower scores, the teams that manufacture 1-2 runs in a game do better in the playoffs than the teams that know that sacrificing will lead to fewer runs over the course of a season.