The success of Die Hard paved the way for plenty of movies with similarly average protagonists stuck in the wrong place at the wrong time (Nicolas Cage in The Rock, Sigourney Weaver in Aliens, even Gordon Freeman in Half-Life), and they all ruled, except for The Rock, which only kinda ruled.
Instead, They're Making ...
The Everyman Action Hero is dead. He was killed and replaced by Jason Bourne, and I don't know if we'll ever get him back. Audiences, apparently, no longer want to watch a good, average cop who happens to be caught in the wrong place at the wrong time; they want to see impossibly tough/smart/resourceful/cool supercops. They want Jason Bourne, or that other guy who isn't Jason Bourne, but is still mostly Jason Bourne.
Just look at the movies that have been coming out. The whole Harry Potter franchise is based around Harry being the special chosen one. The biggest films of the year have been superhero movies. Skyfall featured maybe our Jason Bourniest Bond ever. The Hunger Games is about another "chosen one," a girl who happens to be the greatest archer alive and supernaturally great at surviving. Django Unchained's Django was immediately the fastest gun in the West. And so on and so on. We don't want our heroes to be good; we want them to be the best. Hell, even McClane in Live Free or Die Hard (and, based on the trailers, the upcoming fifth installment of the Die Hard franchise) is more Jason Bourne than John McClane.
Even our Die Hards aren't Die Hards anymore!
I'm sure this is just a trend that will pass and we'll get back to movies with Everyman Heroes just as soon as the world gets sick of Jason Bourne; I just hope it happens before they can do a gritty reboot of Fletch starring Jason Statham as a Fletch who can do back flips and punch through walls.