I strongly disagree. Browsers deal with a hostile environment that poses countless threats to their users, and need to be safe. Arguing that browsers should be minimal and not protect privacy is like arguing that cars should be minimal and not have seat belts.
There is an argument that ensuring privacy in DNS could be done outside the browser. I think HTTPS is a good precedent for putting privacy in the scope of the browser; the browser should attempt to ensure that privacy expected by the user is established or it should refuse to operate.
I disagree with the solution of trusting Cloudflare, but privacy should be considered crucial to user safety in modern browser design decisions.