Why do controllers exist?

Agreed, the name “controller” is what was inherited. Implementation details vary greatly between technologies.

That Ember controllers do not resemble what you saw in “MVC Frameworks” doesn’t make any more valid or invalid either of them.

What is a real controller??? From the link I posted, you can see that there are at least 3 very different definitions of controller. My guess is that you are trying to fit the design patterns used in Ember into what you already know about “MVC Frameworks”, but probably you mean Server rendered frameworks. It will be impossible to make that type of comparison, two completely different scenarios.

Check this video: GoGaRuCo 2013 - A tale of two MVC’s by Yehuda Katz

Also, client MVC is actually closer to classical MVC than server MVC is.