Counterpoint: External CSS stylesheets can be cached, saving future requests, but inlined CSS must be re-downloaded each time and you lose all the benefit of caching the styles independently from the document. Depending on the size of the CSS, there's a tradeoff.
> At my house, an HTTP request over cellular takes 2 seconds.
Here's a nickel, kid. Get yourself a better internet connection and stop complaining about external CSS (because that isn't your problem).
He has a valid point and a valid method of fixing the problem.
Here's a nickel, kid. Go read about the other parts of the world, where people aren't as coddled, and a 2-second HTTP request isn't astonishingly slow.