A VPN and a privacy-focused browser have similar practical usefulness to a private search engine. They cannot be used to create a private search engine.
You don't need a whole browser for that, just a VPN. And that'd likely get their servers blocked for their users if Google's cracking down on them already.
Mullvad sell a VPN and privacy-focused browser so how are they unable to proxy the searches themselves? They already have the needed tools developed.