I hate organic webs. Doesn't make sense genetically. Man made were also the original source, hence their use in the "Amazing" series.
I mean I just remembered that between Avengers and Thor it has been 2 years. Several months for Cap and Stark. Point? Spidey could exist and didn't need any references from other films. So at one point it could have been that post-Avengers, Spidey decided to step it up. No origin, more experienced, we good to go.
As for previous films, I think directors and actors were all perfect, it is more so what Sony creative team and producers forcing to fruition. Spidey 3 would be great if they stuck with Sandman and same if ASM stuck with a villain but focused more on that opposed to romance. Don't get me wrong, I love the love story of Gwen and Pete (read Blue by Loeb and you're done) and I LOVE LOVE LOVVEEEE Emma Stone, but I paid for Spidey and action. If I want love story I'll pop in a Sparks film. That's just me though.