According to the special features on Amazing Spider-Man, it was Raimi's idea that to tell the most interesting parts of Peter's life they had to backtrack to him in high school. Now admittedly, considering the source we should take that with a grain of salt, but there isn't anything yet that has said otherwise. He didn't want to stick around and put out a underwhelming movie. And from what I've read of SM4, it is the only Spider-Man movie so far to really make me want to put my head in my hands.
I don't think they were deliberately trying to swoop those other films tone. The Batman Begins influence is debatable I'll give you that, but to say they were trying to copy Avengers?
Avengers is the Marvel comics tone. This is a movie based on a Marvel comics character. The tone is more on par for Spidey in this movie than the last one, and I don't think Avengers has anything to do with that.
The Avengers influence on the entire franchise is more valid, but it didn't over take this whole movie. We have 1 scene and the end credits which establish a little bit of world building, giving us an idea of the future of the series. And it works for Spider-Man, it is something that was missing last time around no matter how you look at it. And the movies will still be isolated to Spider-Man's world. It isn't like they are digging through Spider-Man's associates to spin off Spider-Woman, or Alpha, or Cloak and Dagger. They are setting up his villains for the future of Spider-Man's own films.