In works of fiction there has to be a suspension of disbelief on the part of the audience -- you're trying to get into a yarn about intergalactic space battles and fights with laser swords in a place and time far away and long ago after all. You have to decide to not disbelieve some things, but these things have to be plausible. The notion that a professional bounty hunter, with his quarry dead to rights seated three feet away, could fire first from a perfect rest and miss by a foot is ignorant, and the thought that anybody who could get gun in hand before that happened would let him do so is even worse. That's not fiction. It's just foolish.
To me, it's worse than the replacement of guns with walkie-talkies in ET.
They might as well have done this with it:
The most aggravating part is, I really can't see how they think the change makes the story any better. He still shoots Greedo stone dead, and after it was clear Greedo wanted to do the same to him. Shooting first didn't make paint Solo as a murderer, just a non-sucker. Like Tuco:
If the scene offended them that much, they'd have done better to cut it from the movie altogether.