Like C on embedded vs C++.
On Windows large majority of apps are in .NET with some C++.
On OS X and derived systems it is all about Objective-C and Swift for desktop apps. C++ is mostly used for drivers, LLVM tooling and Metal shaders.
Android is Java with some C++.
ChromeOS is all about JavaScript.
So no, C and C++ have lost the desktop, nowadays they are used for the graphics composition engine, while the remaining APIs are built on top of it with another set of languages.
If Rust is supposed to beat C and C++ on the desktop, it needs to be able to integrate into the IDE tooling alongside the mixed language tooling experience.
Sure, this is all a matter of 'using it right', but from my experience many developers in the commercial embedded space are not programmers by trade and this learning those intricacies is a real hurdle for them.
And for another point of view of someone more relevant than my insignificant self,
CppCon 2016: Dan Saks “extern c: Talking to C Programmers about C++”
https://www.youtube.com/watch?v=D7Sd8A6_fYU
"Embedded Development with Dan Saks"
http://cppcast.com/2016/10/dan-saks/
Thankfully there are industry certification standards like AUTOSAR, which now require C++, so those devs can choose to keep grasping to C and switch industry or learn more modern languages.
There's actually a website dedicated to tracking this - http://arewegameyet.com/
I think to be on pair with C++, a Rust based engine needs to have some Unreal like tooling.
Which actually seems to be a couple of WIP on that website, so I will now get to track their progress. :)
So it would be interesting to see what Seed Studios will come up with.
Does it? Existing project use what they have chosen long time ago, we can only discuss choice for new ones. And new OSes often choose something else. I am not aware of any new webbrowser, so can't say anything about that.
Rust is really only competing with C/C++, not JavaScript or even Java (or Go).