Error: No Entity Framework provider found for the ADO.NET provider with invariant name ‘System.Data.SqlClient’

I had this problem in a situation where I have a model project that has the references to both EntityFramework and the .SqlServer assemblies, and a separate UI project that uses ASP.NET MVC. I wanted to upgrade from EF 4.1 to 6.1.
I actually had to do part of what was described here: http://robsneuron.blogspot.com/2013/11/entity-framework-upgrade-to-6.html. I want to emphasize that I did not add a reference to these projects to my UI project nor did I add the configuration to the UI project’s web.config, as those steps would violate my separation of concerns.

What I did do was in my model project, I had to flip the “Copy Local” reference settings for EntityFramework.SqlServer (and the EntityFramework reference, to be safe) to “False” and save all, to get the project to put the <Private> node into the .csproj file, and then set it back to “True” and save again, so that True ends up as the final value.

I also had to add the hack line to my DbContext-derived class in its constructor to force the use of the assembly, even though the line does nothing.
Both of these steps I learned from the blog post.

public MyContext : DbContext
{
    public MyContext() : base("name=MyContext")
    {
        // the terrible hack
        var ensureDLLIsCopied = 
                System.Data.Entity.SqlServer.SqlProviderServices.Instance;   
    }

I want to thank the author of that blog that I referenced, as well as all the people that asked and answered the questions on SO, to try to help us get past this terrible bug.