As seen in the image, the nested button is what is missing from my solution explorer. When I try creating a asp.net core web app model view controller mvc project I am not seeing the appropriate views and things are missing from the drop down menu.
What it should look like:
Also this is what I should see and I'm not:


Since you said ASP.NET MVC and not ASP.NET Core, this icon will not display according to the documentation:
Your project(s) need to be targeting ASP.NET Core (or with more recent versions of Visual Studio, .NET as its called now).