It's so sad. We're so trapped. Fortunately, Google (or any other) can't completely alienate a huge part of the web. So eventhough they control the browser, they don't control its content...
The way this will work will be similar to any project with the complexity and funds like the Linux Kernel Project: Recurring sponsorship funds in around $1M+ per month with 1,000 core developers and 10,000 external active developers at other companies also using this Rust browser.
This can also be turned into a Rust consultancy offering their services and expertise around security and Rust which also funds the development of this "Rust Browser" marketed as "more secure than Chrome".
That whole idea sounds almost unrealistic if starting from scratch, but it has worked for the Linux Kernel Project and Red Hat. It sounds more possible if it were spun out of an existing large company. But would it be open-source? I don't think so.