None of this contradicts what I said. When I say "great playoff run," I'm obviously not talking about those that stop short of the finals. This doesn't take away from the fact that Quick played well in 2013, but you have to look at the big picture.
I get that you're a Kings fan and have an emotional stake in this issue. You've made that very clear. Maybe just move on.
I'm a Kings fan but I'm not upset or emotionally invested in trying to prove you wrong, just as a preface for the following RE: 2014 since your take isn't an exclusive one.
The Kings played a real murderer's row just to make the SCF. Any of the three teams they played could have been picked to win the whole thing and there wouldn't be a huge argument against that prediction. Yes, he got lit up the first two games of the SJ series, but then he went out and won four straight elimination games, including a Game 7 on the road. Allows three goals over the last three games combined.
Ducks series, he wins two elimination games that includes another Game 7 on the road. Allows three goals in the two elimination games.
Chicago in the WCF that everyone said was the "real" SCF. Numbers aren't insane but that Hawks team was insane. Goes into Chicago and wins Game 7. No goals allowed in the 3rd or, obviously, OT.
Rangers get dismissed since the "real" SCF already happened, Still, doesn't allow a goal against in the 3rd period or OT of any game in the series, Game 3 in NY shutout and his only loss was a game he gave up only two goals. Game 1 went to OT, Games 2 and 5 went to 2 OTs.
All this to say that narrative matters, which is a big reason why Williams won the Smythe that year. Quick already had the "best big-game goalie" thing going for him after the 2012 performance and then another great showing in 2013. Some sort of fancy stat can say the Kings won in spite of him in 2014, but that ignores the actual hockey being played and is completely false. Winning seven elimination games with four of them being on the road against elite competition is a big deal. Tack it on to the American records he holds--narrative again--and I think he gets in pretty easily.
While being exciting doesn't get you in the HHOF, it doesn't hurt that he was also immensely entertaining.
As for the OP's question: winning three awards that say you are the best at your position kind of makes you a lock, even if the team awards elude you. That said, I think they both get in.