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.
I'd say that 10 seconds of one throwaway scene "fundamentally changing his character" more than the entire rest of the 3 movies he's in is totally hyperbolic, but even your post seems to imply it's an overstatement.
It changes him from Chaotic Good-ish to Chaotic Good? That makes no difference to me.