I've read every Injustice comic so far and also played through the game.
I don't think that Superman was wrong at all with his approach after Joker did what he did, but then the Injustice Superman got annoying especially in the most recent issue, where he Heat Vision'ed through Black Canarys belly, even though she was pregnant.
It's just too much random violence for the shock value, which was fresh at the beginning and here and there very well justified like for example Alfred beating Superman into a pulp, but so many other instances were absolutely not needed like Superman killing Green Arrow, breaking Batman's back and now most recently killing Black Canary's unborn child.
All of that continued in the game, where he Heat Vision'ed through Shazam's skull and broke Lex Luthor's neck (who for the convenience of the already thing plot, was the only guy in the universe, whose morals were totally the opposite... Lex Luthor was straight up a good guy, the entire time and friends with Superman in that universe).
So Superman's idea to enforce freedom is a good one and would make for a great story, but his insanity is taking it too far and I feel that the comic itself is leaving too many things out, which would make for interesting dialogues, like for example the one time where Superman stopped Batman from killing the Joker, after Joker killed Jason. Superman said that Batman had to be the better man and all that shit, but now where something bad happened to Superman he killed Joker and didn't listen to any reason at all and is overall taking it way too far.
That "taking it to far" also made him take the fight versus the Green Lanterns to Earth to use poppulation as human shields.
At this point the Injustice Superman is straight up a madman, who kinda lost track of his original ideas and intentions, when he started enforcing the freedom.
If you want to read a good story about heroes killing bad guys, then read the first ABSOLUTION volume from the publisher Avatar Press. It's very gory and so on, but is in my opinion the best portrayal of a vigilante, who starts killing his opponents.
Injustice is not the prime example of a comic that follows the same idea.