On the surface, the archer hero is simply Batman with a more limited/specialized skill set. And indeed, that's how Green Arrow was originally played, complete with an Arrowmobile, a kid sidekick and an international Green Arrow club of heroes. TV's Arrow isn't too far from the dark night avenger either. But the archer is, and should be, a different animal altogether, even if his motivation may mirror the Bat's, or whether he uses non-lethal "trick" arrows (the equivalent of a utility belt) or the sharp, deadly kind.
Accuracy is what distinguishes the archer from other hero types. The Batman is accurate with his weaponry, of course, but the archer hero is superhumanly accurate. He (or she, I don't mean to slight TV's Speedy, or Arrowette, or Shado, or any other female archer) is allowed to do the impossible when it comes to his weapon of choice, and often relishes doing so. We respond to the incredible focus such a character must have because most of us are just a little obsessed with accuracy, aren't we? Whether it be trying to win a giant teddy bear at a fair or just throwing paper balls into a waste paper basket, we've all experienced the slight agony of the miss, and the sweet exhilaration of the bullseye. Many sports have an element of accuracy - goals and nets and inside certain lines - and many games do too. Imagine being Hawkeye: How much would you rock at pool or darts?
Highly-trained humans acting as superheroes aren't out of our grasp, but a generalist like the Batman might be. But pick up a bow and practice enough, you could be Green Arrow. Well, that name's taken, but perhaps another color? And though we might never reach that level of skill, it's still recognizable SKILL, and like watching any athlete or circus performer, you can't help but admire that very special trick shot. If I again return to Batman (he's the baseline "human" hero, after all), detective work, stealth or even punching as shown in most comics aren't really spectator sports. But it takes showmanship to be an archery hero. And we appreciate showmanship, that we do.