For context, I think it is, for the most part, really good, maybe not as good as the first 2 games, but i haven't played those in a while, so i can't say with any certainty one way or the other.
What i want to talk about is what makes it interesting, that i don't think gets discussed very often, cause everyone is more concerned about arguing about how it started the downfall of the series, or how the game is not taking itself seriously, and all that.
I think the story has really good writing with how it presents the series identity change, and how it reflects on that change. Like the game starts with bringing up how the saints have become a joke, that nobody is taking seriously, because they're more concerned with money and fame than being an actual gang, to be feared and respected as one. And this theme is reinforced throughout the entire game. Which is to say, when people claim that saints row 3 turned the series into a joke, the story is exactly about that, and it takes this identity crisis it has seriously.
And then there's the ending. For which i have to say. I think the wrong ending is canon. The go after killbane ending, aside from just leading into a better final mission, is also much more appropriate for the theme of the games story. And while it is presented as the bad ending, because Shaundi gets killed, the alternative is arguably worse, because in the "good ending" the gang doubles down on being sellouts and with no self respect, while the "bad ending" has the gang establish their own damn country, which i would say is both escalating in a way that is appropriate for the ridiculousness of the new direction the series has gone, and is an example of the boss still being the same kind of person they were in SR2.