Godot 3.x supports C# on the web because it uses Mono.

Godot 4.x migrated to CoreCLR since Mono is a dead end, but Microsoft insists on .NET being the entrypoint in a WASM build. MS initially promised support for .NET being invoked by something else but dropped the feature, leaving Godot stranded. The current proposal is to make Godot a library (libgodot) invoked by .NET.