I've never heard of arachnophobia being a fear of webs rather than, you know, the actual arachnids. It's utterly bizarre to me that that would be GP's first thought...
Right, but even if that was how phobias work, which to my understanding it's not, it still depends on being afraid of the spider to be afraid of the web, while the poster above is implying there are some people who are primarily/only afraid of the web.