The MJ and Miles segments are important for characterization of Peter's allies, but especially because they allow you to feel menaced by the situations the character's are in from the perspective of a mostly-powerless bystander. You couldn't get the same feeling controlling Peter unless he was gimped temporarily by the narrative, somehow, and even then it'd be more frustrating than effective.
The tension of the scenes works because it builds up the villains as actual threats the way random citizen would see them, rather than challenges, rivals or "speed bumps" the way a superpowered individual like Peter does. Peter's "threats" are very rarely about what will happen to him, it's more about what will happen to the city and the people who happen to get caught up in the chaos of the moment.