A long time ago, the possibility of using Java or C# in Gecko (the core of Firefox) was pondered.

Java was rejected because of the huge memory requirements and the unpredictable (and sometimes lengthy) garbage-collection pauses.

C# was rejected because (at the time) it was too tied to the Microsoft ecosystem and there was no way to get it to build on all the platforms for which Firefox is available. I don't remember garbage-collection pauses being discussed, but they would also be an issue.

I think of browsers these days on par with OSes. I mean, they provide a runtime to execute binary code (wasm). They do process management and scheduling. They do a lot of things which up until 15 years ago, we thought bongs to the realm of Operating Systems.

And history has shown that when you need to do that kind of low level code, it's nigh on impossible to achieve acceptable results with a garbage collected language. Many people tried, none really succeeded.

Hence why Rust was made

It seems to me that both C#/Java have build their own niches and are hard to impossible to realistically use outside of them, such as to write a web browser.