Your cloning the track is how I do it. The only step you missed was to open each clone in turn and delete all but the one part you'll keep per track.
Clicking on the keyboard will highlight all the notes of each part and then delete.
Dragging the pointer down the keyboard from the top holding the mouse and stop before you get to the kick drum as example of how quick it can be.
Then rename each new track with what you left behind.
I don't break it in to each part, just groups, Kick/Snare/Hats and Cymb and Toms.