The analytics guys concede that one of the problems with determining whether clutch hitters exist is defining exactly what clutch means. That said, I am pretty confident in saying that I bet these guys have considered the issues that you bring up. That's what they do for a living. If we're discussing these things and we're just mere amateurs, I imagine that they've looked into them. As I've said before, they've sliced this idea up pretty much every way you can think of, and the findings are the same - clutch hitters do not exist.
This quote sums it up nicely:
"Producing wins at the plate is about 70 percent a matter of overall hitting ability, 28 percent dumb luck, and perhaps 2 percent clutch- or situational-hitting skill."