I think he means there are different aspects to web design. You mention in your post that when you tried to design a site well you "ended up with a totally hideous website, using jarring colors, and too many fonts, with really thick borders and dense type." That's not all there is to design.

For example I find I'm not very good at designing something that looks nice but I can design a site/app to work well for the user. The UX is good. Steve Jobs famously said "Design is not just what it looks like and feels like. Design is how it works." You and I may not be good at the aesthetic part of software design but that doesn't mean we should dismiss it all together. Software developers can still be good at 'how it works' and it's possible they might be better positioned than a UI designer to do this.