I agree with you on most accounts, but would changing the game to five faces (Rock, Paper, Scissors, Lizard, Spock) actually solve the problem? Each choice now beats two other choices and is beaten by two other choices, so the player takes damage in 40% (2/5) of the cases rather than 33% (1/3). That is a bit more punishing, but also a lot more complex. Not only does the player need to visualize the 3D dice movement in their head, but they also need to consider the rules of a game they might not be as familiar to as the widespread Rock, Paper, Scissors.
I'd say your other recommendation of making draws hurt the player (maybe a bit less than losses) achieves the goal in a much simpler fashion.