I think ideally, your browser should have a default serif, sans-serif, and monospace font, and websites should default to using it. I think there are useful cases for each style, but we really need not download seven fonts every time we load a website.
When I was taught design in the early oughts they said serif for print and sans for screens. I think it had to do with rendering at lower resolutions and small screen sizes.
Indeed, if you've ever had the chance to compare serifs on a 96dpi and a 170dpi monitor, or even 300dpi, it's absolutely incredible. At those resolutions serifs are absolutely awesome - but I'll never use them at lower resolutions again.