Because Leela (like fastchess mentioned above) has two parts: A neural network predicting good moves, and a tree search exploring the moves suggested and evaluating the resulting positions (with a second net).
If the prediction (policy) net had a 100% accuracy, you wouldn't need the tree search part at all.
One addition: The second part can be run for an arbitrary amount of time, gradually improving the quality of the returned move.
The 60% figure comes from the training games, which are played very quickly, and so don't have a lot of time for refining, thus increasing the prediction accuracy.
In real games, tcec-chess.com/ this "self accuracy" would probably be a bit lower.
It's not non-determinism, it's partial information. The NN part guesses the best move that will be found by search X% of the time. If you just ditched the search part, Leela would be faster and lose out on (1-X)% of the better moves.