Firefox has some Rust components but it isn't written in Rust overall. Servo is written in Rust but it isn't a full browser.
Moreover, Servo aims to be embeddable (there are some working examples already), which is where other non-Chrome/ium browsers are failing (and Firefox too). Thanks to this it has much better chance at wider adoption and actually spawning multiple browsers.
Alas not nearly as modularized as it could be. I think it's mainly just Stylo and WebRender (the components that got pulled into Firefox), and html5ever (the HTML parser) that are externally consumable.
Text and layout support are two things that could easily be ecosystem modules but aren't seemingly (from my perspective) because the ambition to be modular has been lost.