No need to be sarcastic. Disagreeing on a particular concrete situation in which emotional fallibility might play a role doesn't mean either side is ignorant that emotion plays a role in human life. I actually know quite a bit about cognitive science (hence my username, incidentally).
Throughout I mean "mental strength" as in control of your emotions and the mental endurance to apply yourself fully in any situation. I don't mean all mental skills which, collectively, obviously play a huge role in goaltending.
NHL players have faced pressure their entire careers, being closely evaluated from the age of about eight to see if they can make the next teir. Someone who plays badly under pressure likely wouldn't make the NHL in the first place.
Goalies who make "key saves" mysteriously lose the ability to do so whenever one of them leaves their great team.
Studies looking specifically at close, important games have shown the save percentage of the goalies looked at doesn't change (other than going up for everyone proportionately). Perhaps goalies not important enough to be chosen did, but this definitely isn't the case with Hasek or Joseph and several other examples, and that leads me to assume it isn't the case most of the time.
Too much of hockey is important for this to really make a significant save percentage difference and bridge the gap between any goalie that has a significant lead. 72% of the game is played with the game either tied or within one goal, to speak nothing of two and three goal leads which can often be overcome. Because hockey is a low scoring, volatile game, almost any save is "big" if it looks like there was any chance of it going in. It's a small minority of games where any particular goal probably won't make a difference. For those to weight enough to come close to bridging the gap, Brodeur and Fuhr and Osgood would have to basically stop playing everytime their team was down by three.
The effects of your two points cancel out. While a mentally strong goalie would be better in close games, he'd also be better in games that look like they're already decided as it'd be tempting for a mentally weak person to just take his mind out of it because it doesn't make a difference anyway. So presumably mental strength would "inflate" your save percentage with less important saves just as much as mental weakness. I.e., while mental strength definitely helps, the effect would be constant among the importance of shots.
Everything in science, my subjective experiences, my own anecdotal observation, and what I've heard from those people, suggests that someone talented at something zones out (out of other thoughts, into what they're doing) in a lucid way when they're doing well. What this means is that the mentally weak goalies should be more resistant in important games that are close because this effect means they aren't thinking about anything that could distract them, just what they're doing. What's more likely is that a mentally weak goalies are bad not in overtime of game seven, but after giving up goals and going behind early in important games and becoming frustrated which would have a save percentage neutral effect.
Ironically, our anecdotal examples of goalies who can't handle pressure are actually them being scored on a lot in blowouts in important games - which should deflate their save percentage! Presumably other-universe Luongo who handles pressure well would have let in 4 goals both of those games, not 7 and 6, and would have a higher save percentage despite not providing his teams with any wins. The goalies who let in single important goals in important games are the ones whose save percentage would be inflated. So, who does that give us? Ryan Miller? Brodeur - who was in his career well below .500 in playoff overtime games?
A mentally tough goalie with heart and clutchness and all that stuff certainly would have no reason to take it easy during insignificant games, as he wants to give his team confidence by keeping it afloat and has good values and worth ethic. This difference should presumably be made entirely by the mentally weak goalies playing worse during important times. That changes the percentage of game-minutes which are important enough to be played well from 72% to 84% and leaves us with, if we're comparing two goalies one clutch and one talented but mentally weak, 14% of their combined time being made up of what's allegedly throwing off the difference in their save percentage.
Lastly, there's key saves within a game and saves in key games and the second makes a lot more sense than the first for various reasons. For one, it's not overtime or the third period that's more important, it's one goal games. A goalie playing well in overtime couldn't, if he had the opportunity, borrow some of his goodness from the first period and bring it to overtime where it's more important, because just one more regulation goal against and his team loses before it has the chance to get to overtime. Secondly, which saves we remember as key is entirely determined by whether the team scored enough for the save to make a difference and whether that goalie saved enough to keep the team in the game. No one would remember when a goalie made the last save of the game if his team didn't score enough for it to make a difference, and no one remembers the key saves of a goalie that played so well his team was up 3 - 0, not 3 - 2, so the last save of the game that he made wasn't as important.
I do believe playing well in key games almost certainly exists even though it's overstated, but that people currently base this estimation entirely on the fact that a goalie had team success, and it doesn't necessarily correlate with that. I also believe some goalies do not play well when they get frustrated (this is likely the case with Luongo) and so could be worse if their team goes down early in an important game, but that the effect harms their save percentage as much as being good in less important games inflates it, so this isn't a knock on save percentage and can't be used to make "key saves" adjust a "team goalie's" save percentage.
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