I beat the first two Saints Row games a while back, around when they first came out.
Saints Row was basically just a GTA clone. It got popular not for being different, but because it was competently made and fun. Saints Row 2 intentionally tried to provide an alternative to GTA IV's switch to darkness and realism by making the first game bigger, sticking with the crude humor and sex puns, and making everything colorful and over-the-top. It was still competently made and attracted people who were turned off by Grand Theft Auto going in a more realistic direction.
Saints Row 3 has the thinking of "Making it over-the-top and random got us money, so let's go even farther!" Hence the random gimp rickshaw chase that literally lasts not even 2 minutes, a homie that's a black pimp dude and BDSM enthusiast that constantly speaks through Autotune, immediate access to rocket launchers and attack helicopters with unlimited rockets and cannon ammo from the beginning of the game, etc. It has no real focus beyond "MAKE IT CRAZY"
Saints Row 4 is just going straight to aliens and superpowers.