Whether you classify C# as a scripting language isn't really important though. In Unity and other engines C# is the 'language you write game scripts with', which has nothing to do with whether C# falls into the woefully vague and borderline useless category of "scripting language".

    $CSharpCode = @"
    using System;
    
    public class HelloWorld
    {
        public static void SayHello()
        {
            Console.WriteLine("Hello, World!");
        }
    }
    "@

    Add-Type -TypeDefinition $CSharpCode -Language CSharp
    [HelloWorld]::SayHello()

There's your scriptable-in-PowerShell C# ;-)

I agree with you but maybe for different reasons. (Depends on how you defined embeddable I suppose). The primary purpose of a scripting language for gamedev is for mod support. If end users can't open the scripts up in notepad, play around and have them change the game, you're adding friction to this.

I think for the purposes of a third-party system, the ease of embedding is pretty much a question of whether or not it’s already embedded.

I mean, Unity and Godot used to embed Mono... And C# is used as a scripting language for both, in the sense that the user uses C# to call engine functions (dunno about all of Unity's internals but Godot is written entirely in C++) to script gameplay behaviour. It doesn't matter what you classify C# as because they use it for scripting.

There's 3rd party C# scripting plugins for Unreal.