Even Solo: A Star Wars Story, which takes place approximately 10 years before, contained a deliberate reference to the controversy by having Alden Ehrenreich’s Han shoot Woody Harrelson’s Tobias Beckett before he can move against him.
Official canon may deem that Greedo tried and failed to kill Han Solo before his own demise, but it’s unquestionably among the many changes to the Original Trilogy that Star Wars fans have criticized over the years.
George Lucas, however, never wanted Han to look like a ruthless killer.
The main argument against this is that Han was morally ambiguous when we met him in A New Hope, so it makes sense that at that point in his life, before he starts evolving into a more heroic character, he would fire the first shot.