Here's the problem.
Some websites give you a mobile view by looking at the browser, so in this case they see mobile safari and give you a mobile view. This is how the "Request Desktop Site" toggle works, they make it look like it's coming from desktop safari and bam, now you have the desktop version.
Lately however, sites are starting to use responsive designs, which means that the browser doesn't matter, only the resolution of the device. The site then responds to the size. You can simulate this on a desktop by going to the site and dragging your browser to be really small. The site should respond accordingly and give you what is basically the mobile version. Really it's just better designed for the screen that it's now showing on. This isn't something that Apple can directly get around by faking the user agent.