The primary objective would be "separation of concerns", as the model, the view and the controller all have distinct responsibilities.
The author of the original Xerox PARC paper states that:
The essential purpose of MVC is to bridge the gap between the human user's mental model and the digital model that exists in the computer.
If unit-testing were the primary objective, one would be able to easily unit-test views. A look at the landscape of unit-testing projects/frameworks would reveal that it is quite contrary to the claim made. One would typically be using integration and functional tests to test the view.
The problem is that the MVC pattern was designed in a system that doesn't really exist anymore. It was invented in Smalltalk at a time when UI libraries did not exist. To make a window dialog you drew all the boxes, highlighted the appropriate squares, made sure that the text you were drawing ended up in the right spot...etc...
Imagine what it would be like to write a dialog app using nothing but one large canvas. That's the world the MVC comes from.
A "view" in this system was a text box and it was a class that was responsible for drawing the box, the text, drawing selected areas, responding to changes in the text, etc...
A "controller" was another class that took mouse events that occured within this box like mouse moving, key down, key up, clicks, etc...and it would decide what happened. Should we change the text? Should we change the selection? Stuff like that.
A "model" was yet another class that represented the basic data and state of the component. A text box model would have the text of course, the font, selection, etc...
As you can see, in a situation like this the three components are very entangled in the representation of a single idea. It makes sense in this context to speak of a "triad".
Today, if you're working on creating a UI library and using raw drawing commands you might do something similar. But the application of the "MVC" pattern has spread beyond its initial purpose. Now days you have a "view" that may actually be a complete dialog, and a controller that's responding to events like "textChanged" or "buttonClicked". The model in today's MVC is normally something fairly disconnected from the system (but generally linked to the view by providing an observer interface of some sort) and there may be many views associated with the one model.
In a system I recently architected for example we had around 10+ views all observing a single document "holder" and its active document. A main drawing interface interacted with the layout of the document, various property views that observed the selected item and provided a record interface, and a smaller scale representation of the main view that showed the entire document instead of just the visible window. Some of these views had controllers of varying complexity that turned GUI events into changes to the document, which would in turn notify its various views.
Can you still call such a relationship a "triad"? Perhaps, but I think it implies too much of the former, older application of MVC.
Could you share controllers with different views? Depends on how similar the views are. I've found that generally speaking this type of object has behavior to specific to the view it's controlling AND the model it is manipulating to be very reusable...but there's always exceptions.
Best Answer
Using your example the Controller would be what decided what was a legal move or not. The Controller would let the view know how to arrange the pieces on the board at start up using the information it received from the Model. There are more things that can be handled by the Controller but the key is to think about Business Logic on that layer.
There are times when all the Controller does is pass information back and forth, like a sign up page. Other times the Controller is the difficult part of the development because there are many things that need to be done at that layer like enforcing rules or doing complicated math for example. Don't forget the Controller!