May be RMS is wrong, but something on the article is true: we trust too much data to others, data that we could keep secure, and we don't know what they do with our data.
He cares about whether your application code generating responses on the server is GPLed, along with a few thousand other people. He also passionately cares the javascript running in your browser is GPLed, and is perhaps completely alone in this.
It may be simply because he has never used a rich web app, out of abject fear. He normally surfs the web via emacs+gnus+smtp+aliases+wget -- he is not one of us, you can't argue with him on your own terms.