I think that is conflating two separate things - the human stuff is input ... humans play these games inside of our TV sets. I would never dispute the importance therein - but the output is measurable stuff, and the output is what you get paid for. Now Price's postseason output has stunk - now whether that is actually a matter of pressure or just an issue with good teams are distinct diagnoses. (the latter one, while alarming, is much more credible) After all, Price went 10-2 down the stretch as a deadline gunner for the Jays in wins they had to have - and Price was at his best this season for us down the stretch when the games mattered most for securing the division.
WAR is a sum of measurable baseball output - none of the stuff in there is subjective (aside from the subjectivity of its inclusion or exclusion). Now there are different ways to separate a pitcher's contribution to a team accomplishment (run prevention) which is what differentiates the measurements. (fWAR is better for predicting the future, bWAR is better for awards)
There is no doubt pressure can affect people - at the same time, those factors matter less when dealing with a profession you have been doing since you were a child - especially if you do it well. And either way, it should come out in the wash. Could Price be pressing a bit in the playoffs - yes. But he was pretty good in a lot of big games which were essential to get to the tournament. It's not binary.