Certainly pressure is real. Heart rates go up, focus can be harder to maintain, etc. But the pressure of pitching in the 9th inning compared to other times is largely a construct, created by the save stat and years of baseball tradition (and closer salaries). And if sabermetrics say that almost any pitcher can protect a three-run lead in the 9th inning, then yeah, like julio says, teams and pitchers should understand that. Otherwise, it's largely self-fulfilling.
Certainly there are skills, and knowledge/experience, that can be acquired by being in a familiar situation repeatedly. But it's not magic. And closer don't really "melt down." They just fail, as many relievers do, and we perceive that as a meltdown, and remember the failures more prominently (example - Ryan Madson's blown saves in 2009 when he filled in for Lidge were more memorable than his blown saves in the 2008 World Series).
Here's that SI article about the current trends
In 2015, a record 21 closers saved 30 games or more. That number fell to 16 in ’16 and to 11 in each of the next two years. Meanwhile, the number of pitchers who secured at least one save has increased every year for the past five seasons, hitting a record 165 last season, including 43 with 10 saves or more, another record.
