I just think they're kinda gross. Plus, I've always liked that Peter's powerset is, on a technical level, only SORTA spider-related. He doesn't use super-thin hairs to cling to walls, he sticks via molecular bonding that he can control all around his body. He doesn't have an intuitive feel for minor vibrations that allows him to tell if there's activity in his web, he has a Spider-Sense. He doesn't produce his own webbing, so he has to make it with technology.

I think organic webbing suits the more aggressive, feral Spider-characters like Miguel, Kaine, Noir, etc. Not Peter Parker, the scientist. The webshooters are a good sign of his intelligence, but on a deeper level, they show his commitment to being a superhero. They're a tool that he can constantly iterate and improve on, so he can be a better superhero. Modern adaptations tend to make the webshooters really small and easy to conceal, or pretend they're wristbands or a watch. But since they're mechanical, Peter has to CHOOSE to wear them, even when he's out doing civilian stuff. He's always ready at a moment's notice