I would like to think that the idea of bosses came to mind that since Saints Row 1 was lacking in things like how the deaths in the finale were done from a background angle, (Tanya's death in a cutscene, Angelo's plane explosion, Joseph's truck explosion) it took out the ideal challenge of actually going to hardcore combat. Sure, dealing with Joseph Price was a challenge, but you never actually get to face him in person aside following him and avoiding pipe bombs. Since that was an obvious element lost in the gameplay when fighting the bosses, it seemed like that a good option would be to make it more personal instead.
I find that since the first Saint Row didn't feature any encounters, it didn't create all the tension that the Playa has. It seemed clear that during the Rollerz missions that when Lin had died, the Playa took it with some sadness. He got revenge from Price, but he didn't get a slower death by jumping out of a burning car and getting dealt with in a cut scene. Though since he acted as a mute back then, the point seemed to make sense for his silence. In a way, it all could've come down to his silence that avoided the conflicts back then, however, that wouldn't make a good excuse to have a personal fight.
Saints Row 2 however, he had been more open after a five year coma, which possibly lead to his breaking point of the fear that he had admitted that he had when he explained it during Saints Row IV and turned him into a badass. Since he was actually speaking now, I think adding boss fights made more sense because he had more words to share. I like boss fights when both characters would have a personality to understand, otherwise it seems just like a fight. Nothing else.
So personally, I think it might be either because The Boss grew an attitude, or simply it could just be because it was the sequel to make things about Saints Row look better. An improved world of gameplay to the series.
Sorry for the long paragraphs of words I have given. I like to add my opinion about things like this. I should find a way to shorten what I write, haha. And sorry if this doesn't make sense to you as well, partly.