Hey hey!! Very nice game! Looks clean, responds well, and the sounds and music are wonderful!
I think the one big thing that might have solved the unwinnable state issue is if, when the player rotates a ring, it doesn’t bounce back into its original place, but rather the pieces a moved after the rotation. This would allow players to chain-move pieces to score points. To add difficulty, the player could have a limited number of moves or each move could cost (score)points. This would make the game infinitely solvable, but the strategy would be the player choosing their moves wisely so as to either not use as many moves for a match, or to retain as much of their score as they can.
Just my idea. Otherwise, A very lovely game!! Excellently done!!