It depends on which typescript toolset you use, but generally speaking you're probably riding on mix of C++ and Javascript if you're using VSC. VSC itself uses C++ in it's core components since that is how electron works. Similarly the language server and tooling for both Typescript and Node are build with C++. If you're fancy and use Bun you're running on Zig. Eslint itself runs on Javascript, but the parser it uses feeds it something called an abstract syntax tree, and different parsers will do this differently.

So the relatively simple answer would be no, it would not be faster with C# (or Go which would likely have a similar speed to C#).