Agreed for the most part. There is Servo[0] which is Mozilla's playground for a Rust Browser. It's not really usable for day-to-day browsing, but it's a neat proof of concept. They are actively moving stuff from it into FF, and FF now requires rust to build. Google definitely cares a lot about security. It's great to see them embrace U2F so heavily, that even Chrome supports it out of the box. I'm not sure FF will ever become Rust-only, but if it did, it would take quite a while I think.
[0] https://servo.org/