From what I understand, LSP support for c# isn't very good, and is from third parties, not MS themselves, because they want you to us Visual Studio on Windows.
Microsoft provides C# LSP for VSCode (https://marketplace.visualstudio.com/items?itemName=ms-dotne...).
The problem is that it's closed source, and its license specifically prohibits its use with anything other than VSCode. Unfortunately, this is becoming the usual modus operandi for first-party Microsoft extensions for VSCode for all the talk about how they "love open source".
That said, F# was years ahead of C# in features C# is still chasing, and is driven mostly by the open source community. That community is more in academic and finance areas where Linux-first is common. The language is standardized and plugged into VM improvements over time.
Frankly, I see the lesser degree of entanglement with MS corporate interests as a boon for the language and its ecosystems long-term utility.