Consider your question in the idealized case of zero transaction cost and where the underlying stock price follows geometric Brownian motion with constant volatility -- identical to the implied volatility used to price the option.
If the delta hedge is rebalanced over time short time intervals of length $\Delta t,$ then the cost of hedging is a random variable that converges almost surely to the option price in the limit as $\Delta t \rightarrow 0$. In practice, of course, it is impossible to continuously delta hedge. Furthermore, in the presence of transaction costs the hedging cost with continuous rebalancing diverges to infinity -- Brownian motion has unbounded variation.
Hence, the hedging cost for any realistic strategy has some distribution around the option price with non-zero variance for $\Delta t > 0$. The actual cost will depend on the path of the underlying price. Even if the option expires OTM, there are paths where the cost can be very low -- underlying price runs steadily in the OTM direction, and there are paths where the cost can be very high -- underlying price oscillates frequently through the strike price close to expiration and finishes OTM.
In summary, with delta-hedging over non-zero time steps, the hedging cost (conditioned on the option expiring OTM) is not less than the option price with certainty.