Hm, not a fan of that to be honest. People relate to him being an every man, with his relationships and how he deals with things like every other human would. Giving him organic web-shooters as a way to identify with him is silly.
Sam Raimi's point was that making Pater such a genius that he could invent the webshooters and web fluid on his own makes him less relatable. That's fine for someone like Tony Stark, but Spider-Man has always been a sort of everyman - a "what if it happened
you?" type of character. If he's merely smart (ie, gets good grades, is knowledgeable enough in scientific matters to hobnob with Otto Octavius, etc), that's one thing. But beyond that, the more of a genius you make him, the less relatable he becomes. (Personally, I felt that effect in the Marc Webb movies - Peter Parker's genius was emphasized so much that he ceased to be an everyman character in my view).
Making the webshooters organic causes them to be just another of the powers that were thrust upon him by the spider bite. That again plays into the "what if it happened to you/it could happen to anyone" appeal of the character.
In the organic webshooters scenario, if you or I happened to have been bitten by the spider, we could get Spider-Man's exact set of skills and powers. You or I could be Spider-Man if only we had been bitten by the spider instead of Peter.
But in the mechanical webshooters scenario, if you or I happened to have been bitten by the spider, we would only be able to climb well, jump high, lift heavy weights, give and take hard punches, and have spider-sense. But we couldn't be Spider-Man as we know him, because neither of us (unless you're hiding a big secret) are next-level geniuses that could invent a liquid that can be squirted out into cables that are stronger than steel and dissolve into air after an hour. So we couldn't swing from skyscrapers or web criminals to walls.
Thus, the organic webshooters make Peter more relatable.
If I recall right, there's also a Sam Raimi quote somewhere (although I can't find it now) where he said something to the effect that if Peter was able to invent the webshooters and web fluid, he could patent them and become rich, and the fact that he doesn't do so makes suspension of disbelief harder than with the organic webshooter.
After all, think of all the applications of such technology. The police, fire department, etc, could come up with many great uses for it. It would sell like hotcakes.