Interesting idea for a challenge I'm not sure about how useful it is to prove whether or not Construct can do anything - difficult logic problems are difficult logic problems, no matter what tool you use - unless the tool solves that specific problem for you, which general-purpose tools usually do not. By "Construct can do anything 2D", I think we mean limitations in Construct shouldn't get in your way and prevent you from making the game you want.
In this case, though, Construct does effectively solve the problem for you
<img src="http://www.scirra.com/images/mouseevent.png">
If that's not a perfect example of events being a quick and readable way to do something, then I don't know what is
I've attached a copy where I just added the RTS behavior to the mouse, carefully adjusted the settings, and added an event to move the mouse to the cheese. The RTS movement has a very decent A* pathfinding algorithm built in to it which will basically immediately work out the optimal path to the destination, then move the object along that path. The A* bit is great, but it seems the RTS behavior's code to move the object along the path is a bit ropey. It wasn't designed for small, enclosed layouts like a maze - it's aimed at wide open spaces with occasional obstacles. So I had to tweak the settings a lot, but now the mouse basically walks directly through the maze to the cheese.
It's pretty surprising to see how bad the RTS behavior fared with the default settings, it gets stuck pretty quickly. I guess it needs a redesign if I ever get round to it. The mouse also has to be very small and use the smallest cell size available in the RTS behavior, since the walls of the maze mark entire cells as solid, and the entire map would be marked solid with a large cell size.