Maybe I'm just some weirdo minority Star Wars fan but frankly I've never cared who shot first: one way or the other. Heck it was a while before I even noticed Greedo shot first in the special editions.
The reason people care is that it fundamentally changes Han's character.
If Greedo fires first, Han is just a regular hero, probably Chaotic Good.
If Han fires first, he's somewhat of an anti-hero, more Chaotic Neutral with good leanings.