I absolutely think we can question that. He played harder and better for Team USA than he did for the Red Wings this season. And, yeah, he got hurt, but it was his decision to go. That makes it his problem, and he shouldn't be pointing fingers at anyone else. If he's upset with missing the playoffs, he should look in the mirror first.
And that's not even getting into the mess that the locker room clearly is. It's his job to manage that, and it's gone to shit. He has not been a good captain, and that doesn't seem to bother him very much.
So, yeah, maybe he skates hard, but I wouldn't say he's tried real hard in other respects. He clearly thinks some aspects of his role are actually just someone else's problem. Which explains this "the buck stops everywhere else" type response.
EDIT:
Not to mention, total leadership 101: you don't get to freak out in front of those you are leading, or you will no longer be able to lead them. You have to project a sense of sureness and strength, and Larkin sure as f*** did not do that. He undermined his own authority by virtue of undermining Yzerman's authority. Now everyone is going to wonder what the plan is, and if they should believe in it—especially if Larkin does not. You can't just undo that. Maybe he didn't mean to, but it's done.