Because the Lords were written well. We saw Lex become president and do something so horrendously fucked up the League had to assault the White House and put him down.
It felt less like they wanted to take over and more circumstances led them to take over, sort of like the Gruenwald Squadron Supreme. There was a power vacuum and they filled that vacuum as best they could.
Superman's Regime is retarded because
1. Batwank out the ass. "Only Batman is strong enough not to be corrupted by power. Only Batman is strong enough to not fall for the Joker's mind games". Everyone knows Bruce is the League member closest to the edge and yet he's the only good one.
2. Somehow killing Joker is supposed to be the line that once crossed cannot be stepped over again. Killing Ted Bundy with a nuke somehow leads to global dictatorship. And its all to wank off Batman's no kill rule. They even do it again with Brainiac in Injustice 2. "No Superman we can't kill the genocidal alien we must be good".
3. Superman and Wonder Woman are needlessly cruel and evil. It's not enough they have to take over the world. They have to be evilllllll about it. They have to employ supervillians and kill Captain Marvel for disagreeing with them (there better be a Billy premier skin Boon you fucker)