I actually love the organic web shooters. It makes as much sense as the rest of the powers he endowed from the spider bite and save a heap of time and explaining.
I always thought it was dumb that the signature super power of SPIDER-MAN wasn’t even a super power originally.
IIRC, Lee & Ditko considered making Spidey’s webs organic but ultimately decided against it so as not to make the character “too creepy”.