Yes, sometimes stalemate is the only way to win, if you make your opponent make stalemate.

Is a stalemate supposed to count as a failure? According to the instructions your turn is skipped, and the opponent loses. Consider the simple example of a game starting with 5 cakes

I go first
1 > 3 > 2 - I lose
2 > 1 > 2 - I lose
3 > 1 > 0 > 1 - Stalemate in my favor
You go first
1 > 3 > 1 - I win (So you would not choose this)
2 > 1 > 2 - I win (So you would not choose this)
3 > 1 > 0 > 1 - Stalemate in your favor

In this sceenario I can't win 100% of the time. Even if I choose who goes first my best case scenario is a stalemate in my favor