The irony is that the way Tim Berners-Lee designed the web, the web server was to send you a minimally-structured set of information to display, and it would be up to the client to format it in the best way for the local display. This meant things like font sizes, page flow, in-line photos, etc. should adhere to settings on the browser.
The designers and page layout artists were horrified at this, and did everything in their power to subvert this model and return control of how the site would appear back with themselves. That's why flash websites were so popular in the early 2000s - it gave them complete control of how the site would appear, giving the user none. Gradually they've figured out ways to take away control from the user using regular html, which is why you now have websites where you can't zoom, can't resize fonts, everything is locked to three columns (menu, text, ads) which you can't move, resize, or rearrange, etc.
The way Tim Berners-Lee envisioned the web, there would be no need for a desktop site or a mobile site. You just create one site, and it's up to the visitor's browser to format it in a way which makes it most usable on the display device. The need for different desktop and mobile sites only arises if you design your site so that it will only operate at a certain resolution or screen size.