I read it and I'm still confused so I'll just ask: Is a valid path to the destination space for a piece required or is it literally a case of go anywhere for every move?
Yep, that's exactly right.
- Pick up your piece.
- Remove your opponent’s piece.
- Plop down your piece.
Winning strategy has nothing to do with your pieces' positioning on the chess board -- instead, you're focused on moving to those shaded squares on your game sheet to capture your opponent's pieces -- either to win by capturing all your opponent's pieces first, or to obstruct your opponent by capturing the piece they need to progress on their game sheet.