This one I totally agree with and also brought up. The mechanical webshooter was a service to the comics but the film didn't handle it well. First off, it adds almost nothing to his character because he didn't create the web, only a devise to make it shoot out of. So he will regularly have to steal it from oscorp and they will realize someone is stealing their supply. Can't they just look at their security cameras?
But mostly, like you said, the web in this movie IS organic... it is woven by the genetically altered spiders. So if the spider that bit him could create this steel-like web, why can't Peter if he inherited its powers?
It feels to me like the filmmakers wanted to be closer to the comics but knew a kid creating super-strong silk on his own was too implausible so he just creates the shooters. Most of the reasons that people disliked organics can be placed on this rendition as well. It's closer to the comics, but only in the most superficial reasons. We won't get to see him create different kinds of web tailor made to help him defeat certain villains because he didn't create the web in the first place, he just steals it.