Spider-Man having organic webbing is objectively better than him having to create it himself.

Yeah, the guy who got his powers by being bitten by a radioactive spider, only has one slightly spider-related power. (Climbing walls). Without his webs, he's basically Daredevil with a cooler suit.

There a few reasons why people prefer mechanical, but they are easily refuted.

"It shows how smort his is!! 🧠"

There are a thousand ways to show a character's intelligence without them creating a new compound just to fit some sort of motif. If he can create webs that no one else can recreate, he's smart enough to build a thousand other contraptions. He just really likes spiders, I guess. He could create rocket boots to fly around with, but nope, swinging around is more fun.

"Dramatic tension because he could run out of web canisters!"

You create fecal matter organically, but you still run out and have to recharge. (Excluding all the crap spewing out your mouth). Webs are no different. There can still be dramatic tension when he runs out of webs even if he makes them naturally.

"Wristussy is gross and I don't like it"

Not a serious complaint.

"It was part of his original concept! You can't mess with the OG!"

Things can be improved over time.

Plus, part of Spider-Man's whole message is that "it could be anyone behind the mask". Well, if he has to make his own webbing synthetically, then it could be like, one of twelve super-genius people behind the mask. Not anyone.

It's also extremely immersion breaking that Spider-Man creates his webs while living in a suburban house with his aunt.

There is reasonable middle ground though. Suppose he makes his webs organically, but he invents shooters that allow him to shoot his webs further, more accurately, or use them with other properties, like electrical webs or acid webs or making large web nets or whatnot.

People who want Spider-Man's webs to be purely synthetic and mechanical are just plain wrong.