Yeah, M-V-C are all roles, not concrete objects. The C mediates between the input devices and the model, but in practice views can and often do fulfill that role as well. Cocoa views, for example, also fulfill the C role.
Different formulations of M-V-C have the C deal with more complex interactions, with sequences of interactive prompts like wizards.
The update is essentially automatic, and yes: MVC already solved the "problem with MVC" React/Redux/Elm/SwiftUI claim to solve. In 1979.
I like my Controller to be responsible for all the "business logic" so that its all in one place. It's the important part. The View layer is always fairly verbose and full of fluff. Especially if you have a lot of animation and formatting type code.