The thing is, if you randomly generated numbers for players in such scattered and small samples sizes as career playoff PAs or IP, you'd get a few outliers that do better than the norm. How can we know with any certainty that the numbers by Papi and Schill weren't just the luck of being hot, during those selected performances?
I get the point about facing tougher pitchers (Papi) and batters (Schill), so equal to regular season stats really means they did better. I bring up the Josh beckett example, a lot. The guy was on his way to becoming one of MLBs best playoff pitcher in history, then suddenly, he wasn't anymore, while still in his prime years.
The sample sizes are generally too small and scattered. The deviations from the norm look precisely the same as a randomly generated sample size of players and numbers over such a sample size.
I will say, I'm not sure about the idea of a player just being lucky vs some sort of skill helped by some sort of "inner calmness" or whatever we want to call it. That very well may play a role in improved or worse numbers by some players. I just think it is nearly impossible to prove, and I think the idea or "clutch" is hyped and over-stated.
Look what people were saying about Devers into mid May, this year. The guy has had some mammoth hits for the Sox, over the years. He goes 5-6 weeks hitting badly with men on base, and all of a sudden he is a choke. Nobody is talking that talk, anymore, except maybe in a few game threads..