Why practitioners still use trinomial trees

A one-step trinomial model does not determine option prices uniquely. Why, then, do banks routinely use trinomial trees?

Because they are used as a numerical scheme for a continuous-time model, not as the model itself. The tree's probabilities are chosen to match the drift and variance of an underlying diffusion (Black-Scholes, Hull-White, etc.), and as the steps shrink the price converges to that model's unique price. The extra branch gives flexibility: better stability and convergence, easier fitting of mean-reverting rates, and nodes placed on barriers.