The retcon later introduced in the comics was that the spider bite gave Peter an innate understanding of how spider silk worked, and that gave him the ability to create it artificially. Sure, that's a little silly, but I think it's less silly than having spinnerets conveniently grow in his wrists
, of all places. Besides, it has the advantage of establishing him as an inventor and scientist rather than just some guy with mutant body parts. Peter's ability to solve problems with his intelligence and scientific knowhow has always been an integral part of his character in the comics, and it's one of the things I like best about him.
Besides, as I said, the comics never really committed to the biological-webshooter idea. They did a story that introduced them and then they never followed up on it; most of the stories that followed in the comics would've been no different if he'd still had webshooters.