You're confusing "the web" with "networked software".
When you use the web, you have no control of the server but full control of your client (if your browser is open source). When you use a networked app, you have no control of the server AND no control of the client.
"Do I want control of the software I run on my machine?" is a separate question from "do I want that software communicating with other computers that I don't control?"