Thanks for trying the game out, it is a bit of a rushed mess. The blocks with flat edges are intended to line up with flat tiles, allowing you to reach places that don't have connections. As for the orange block, it does give you the power to move pieces, and later on there is a purple piece that gives you the power to rotate them.
As for the pieces not being movable when you stand on one, it is extremely necessary. For example, from the area with the three green pieces, you could reach any space on the map by continuously swapping pieces.
Edit: I misunderstood the comment. The first puzzle is solvable using only the pieces directly on the path, the second puzzle is supposed to use one from the first. I added a gif of the first area below.