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.
I said this in the other post about this, but here are my thoughts...
My problem isn't because of the cold-blooded thing. I actually disagree with the assertion that it changed the character in regards to his alignment. Han shooting first wasn't a cold-blooded act. Greedo was sitting at the table, with a gun pointed at him, literally telling him he was about to kill him. Han shooting first didn't establish him as cold-blooded. No matter who shot first, it was still Han defending himself.
What the original did establish, however, was that Han was a bad***. In the original, we were watching an old fashioned cowboy shootout.
And Han won that shoot-out.
In the new one, Han actually lost the shootout and only walked out of that cantina alive because Greedo can't shoot someone sitting 2 feet away.
Those few seconds hurt the scene for me because it was 1. incredibly goofy looking, 2. had a character that was such a bad shot, Stormtroopers insult each other by saying, "You're such a Greedo," and 3. Han LOST the shootout; he just killed Greedo because Greedo can't shoot.
It didn't destroy the film for me, but it is definitely one of the only two changes (the other being Vader's "NO") that I think was a mistake to change.
The rest of the changes, I actually liked, more or less.