The great thing about Saints Row: The Third's plot is the way it escalates. It starts off as another revenge story between a couple of over-the-top street gangs and then halfway through, the violence gets so bad that the authorities call in the military, but because the Saints are already established as these unstoppable badasses, it's not just the ordinary military, it's a special unit that wields all sorts of science-fiction weaponry. Once they're introduced, STAG becomes the game's main antagonist, and the story shifts to how the Saints manage to successfully fight the sci-fi military. At that point, they've crossed the line between ordinary villainy and cartoonish supervillainy.
Which is something I didn't even know I wanted until I got it. A lot of games let you play Batman, but generally even crime games stop short of letting you play The Joker.
Although, it's debatable how Joker-esque the Boss is meant to be. If we go by the SR3 characterization, they're both trickster figures with a lethal sense of humor, but the Boss doesn't really indulge in sadism or malice (forgive me if I'm being too hard on the Joker, I only know him from the movies). Rather, the Boss's main sin (aside from some dubious stuff with the sex-trade missions, which, since the implications are glossed over and the Saints are presented as "rescuing" the prostitutes, I'm going to choose to believe is not as gross as it could be) is a careless indifference to human life. The Boss doesn't care about collateral damage or the consequences of their actions. If something looks fun, they do it.
Which would be a pretty chilling characterization in a drama, but in the world of Saints Row . . . The Saints are worshiped, despite their many horrific crimes. "Professor Genki's Super Ethical Reality Climax" is a gameshow where you run around shooting people in mascot costumes for cash and is apparently canonical. Josh Birk is an A-list celebrity that shows no legal or moral trepidation at robbing a bank with the Saints. Even the anti-gang authorities are overzealous fascists who are willing to target civilian populations in the pursuit of victory.
So, I don't know. I sometimes feel conflicted about driving up on the sidewalk and plowing through a bunch of people, but then my character's popularity doesn't suffer from it. I think we're suppose to realize that the worldbuilding is completely bonkers and that it's folly to attach serious moral weight to anything the Saints do (their world doesn't judge them, so why should we?). This is an issue that will come to the fore in Saints Row IV, wherein the Boss completes their transformation to full-on "puckish rogue," but I think they really hit the character's sweet spot in SR3 - they're a menace to society, but society's cool with it.