"The only thing that a website may need to know is your browser version"
They don't even need to know that. No website should be making decisions based on what browser/version is being reported. If the site is doing tricky things, it should test for functionality rather than try to assume it based on make and model.
Otherwise, I'm with you. I really think that browsers should tell webservers pretty much nothing at all. Or, at the very least, allow users to disable such reporting.