Even if you call it managing, it's too easy to see emotions as something with a volume knob you can just turn down whenever you like. But it's more complicated than that. They're basically hardwired into our brains. Some people's brains are wired differently, but even in a completely normal brain (if such a thing exists), there are systems that kick in and do stuff whether you want them to or not. As in parts of the brain that activate and shift gears for other parts of the brain.
So there are some things you can control but also some things you cannot control. Which means to manage them, sometimes you need to understand how the system works and the kinds of changes you can make. It's a bit like fixing a complicated bug: you can't just comment out the code that is doing whatever you don't like. You need to understand the whole system so you can make the right change in the right place.
I think your fear example is a good illustration of this. Experts have figured out how the system works, and they know where a change can be made, and that's exposure.
> The solution is actually to experience more of that fear - in a controlled, gradual and safe manner, which is what exposure therapy is all about.
is true for some of the people some of the time. There is a wide range of approaches that will work for some and not for others, in various circumstances.
Again, I'm not saying that this is what you meant by "controlling" or "moderating" emotions, but just want to clarify that learning how to express emotions in a constructive way usually makes an argument that much more forceful that trying to "rein in" your emotions and make them subservient to logic.