B vs S isn't about Batman's war on crime it's about Batman's war on Superman. I looked it up but why didn't you simply explain when I asked you? A movie can fail for many, many reasons and that's just one reason it fails. Ideological battles are engrossing, it's what makes great stories between protagonists and antagonists. It's a raison why Nolan's Batman trilogy is beloved, every film Bruce's ideology is confronted with its opposite. The problem is Snyder implies rather then talks about the struggles, sometimes he gets into it but then does nothing with it. He's got loads of material to explore in B vs S with ideologies and he fails at truly exploring them where Nolan would cover every inch of it to milk the drama.
Compare this with what Nolan did with Batman Begins:
https://www.youtube.com/watch?v=uiaRYQlsjy4
https://www.youtube.com/watch?v=2J4yJXYPJDw
There are other scenes, but this should be sufficient.
That's surface detail. And Snyder's Superman's idea of "protecting the weak" is by being apathetic about people dying when he rams himself through buildings with super-powered aliens (man of Steel), doesn't care when heavily armed thugs fight Batman like they're in a war zone (B vs S), and doesn't fight crime in the DCEU. This is why he's caught off-guard by Lex, despite the fact he had the people who would killed his mother in his sights if he hadn't simply focused on Batman - it's like an unintentional Uncle Ben moment which Superman never connects. He will put anyone who endangers Lois through walls or threaten to burn them alive with his health vision when they threaten his mother, though. Batman's obsessed with killing Superman because of the Battle of Metropolis.
There is no "forgiven" scenes in those movies, I don't know whee you're getting at. Police are afraid of Batman, while Superman is controversial and that's not gone into much - the Battle of Metropolis isn't bought up aside from the memorial. Snyder brings up society being impacted by Superman and wants us to question it but that goes nowhere because of the court hearing explodes and the montage just ends with "Superman is." He's afraid to put his Superman really under a microscope, except when he's pouting. The films have contradictory messages about his impact and don't go into it as mochas they should and it's retconned entirely by Justice League he may as well be a new character in Whedon's movie. We get the funeral in B vs S, but its unearned. He's hasn't been Superman for long and he's isolated from the public and when he did go to court it got permanently sidelined by the explosion, which he's blamed for. The funeral is like what classic Superman would get after spending years building that trust up.
I could buy Batman being an anarchist, but Superman? No. They don't have enough detail about his political opinions on anything. I did get a conservative/Libertarian vibe from him, with his being raised by these cynical Kent's who can't say rescuing children was right, are paranoid about the government, that they don't "owe" the world anything (the complete opposite of Spider-man's motto), Clark going to a priest for help rather than a therapist or a close friend, he's mainly concerned about protecting his girlfriend and his other by B vs S, and his biggest (temporary) alliance with humanity is with the US military in Man of Steel, there are no sub-plots about Clark's opinion on whether killing people is unjust, and collateral damage is nothing to him.
Batman does harm people, sure, but not all Batmen do that the same. Snyder's Batman isn't the same with violence as Batman TAS. Brutal does not entail dropping cars on people like this one does, that's actively killing people. That it does, however, Snyder Batman is not Arkham Batman. Except this is about a movie, not a video game and characters must be consistent or it looks sloppy. The MCU actively tries to cut down on the destruction, like the evacuation in Age of Ultron - a foreign concept to these two. Its only by accident the pseudo JLA fight Doomsday in an abandoned area because the reception was negative to what Superman did in Man of Steel in a crowded city. It is definitely controllable, Batman's heavily trained and experienced Superman's just a fool who can't bother going to learn martin arts or train with his powers. A big reason Superman in other media is such a strong opponent is that he knows how to wield his powers and fight properly. He's not just brawling blindly against Metallo. Everything Batman does in the DCEU is controlled in his violence and action. Snyder loves the excitement of the destruction and the pretty images, the implications of how horrible it is isn't a factor. Superman should be doing more than just punching and kicking but not this Superman, he doesn't even use his x-ray vision after Man of Steel. His Batman drops cars on people, he doesn't need to punch someone to kill them. How this violence is staged in movies speaks to the audience, and Snyder has a certain aesthetic he sticks too which is ultra violent. Like you said, they go all out. There is no restraint.