First Cano. There is no doubt that his defense is beyond reproach. As a hitter, Cano is like the low cost window cleaning solutions that Windex commercials warn us against: mostly streaks. While his work ethic has at times been called into question, it's mostly been unfair, especially in light of how he's produced over the years. That said, there have been moments where his focus seems to have parted ways with him. He recently failed to cover first base on a bunt attempt, and while his tendency to not run hard out of the box on grounders is generally harmless, occasionally it makes him look bad (e.g., balls that a fielder doesn't handle cleanly and his slow trot gives them time to recover and make the play at first).
No question that he was awful this postseason, but he does not have a history of always choking in the postseason, just of being streaky, as is his wont. That said, the Yankees can't afford another very expensive postseason disappearing act.
Which brings us to A-Rod. A-Rod's capacity for shrinking in the moment is more like a super-power than a tendency. It's like he was bitten by a radioactive Fredo, endowing him with the amazing capacity to do nothing right when it matters.
But I will give A-Rod this, while he can no longer hit, he can still make pitchers throw quality pitches. He doesn't expand the strike zone the way he used to, and so he at least helps contribute to wearing out pitchers, unlike Curtis Granderson, who on average sees -2.3 pitches per at-bat.
The biggest problem with A-Rod at this point is that a history of lower-body injuries more or less has deprived him of the ability to remain effective late into the season, let alone the postseason. This could be easily addressed by limiting A-Rod to DH duties, but that's a very expensive DH. Besides, he still plays a competent third base, but at this point you have to decide whether you'd rather have A-Rod the third baseman or A-Rod the power-hitter.
A-Rod also has a bad tendency to be a distraction, as his recent attempts to pick up women while benched has shown. But if the Yankees had swept the Tigers, no one would care. Sure, it would reinforce a perception we already have of A-Rod, but that's just the point, we already think A-Rod's an intolerable piece of $#!%, so he can't really further mar his reputation with on or off the field antics. All he can do is make us hate him as a player, which, except for 2009, we already do. So he really can't win.
The Yankees should never have extended him the way they did. Cashman knew as much, but it wasn't his call. So we just have to live with A-Rod, and if the Yanks can't trade him, they may have to think seriously about whether they should keep him in the field anymore. Occasional DH days are not enough to keep him healthy. Even with more than a month lost to a broken hand, his lower-body injuries caught up to him, slowing his bat and making it impossible for him to do anything more than accurately identify pitches he cannot catch up to.