Alright, there are 2 sides to this
akwardness and capability
the webshooters are ultimately a way of showing off peter parker's genius ability to manufacture something completely brilliant out of nothing. It also showcases his ability to put the commercial gains of the device to one side and do what he believes is responsible (when quite possibly the money he would get would help his immediate family which could be also deemed as responsible). I think somewhere deep down he knows this but his guilty actions won't let him accept it and he believes the path he has taken is the best one to redemption.
If there were no webshooters, then he would need something to fill this void in the character.
Secondly, they are a plot device, they need refilling, updating, higlighting, evolution. they are also cumbersome and need hiding. So this part of the character would also need some filling with organics. Saying this, I'm all up for the variety in his diet causing differences to his webbing consistency, how full they end up, their additional properties. Infact, he could purposely cut something out of his diet to stop being spiderman or someone could 'poison' his diet to cut a vitamin combo to make the organics naturally (as well as maybe his other powers)
So as long as those character values could be replaced, I really don't give a monkeys whether they are organics or mechanicals. Just don't change them and leave the void. Replace it with something else to re-establish equilibrium and maybe enhance the character.