On top of the many other good answers here, you also have to be careful about distinguishing between the "payoff" and the risk exposure.
Let's assume there are no jumps; that zero-cost instant frictionless trading were possible; and none of the classic shortcomings of the Black-Scholes assumptions apply.
Pick any option you like: long or short, call or put, any instrument, any strike, any date. How do you replicate the payoff in the scenario that the market simply doesn't move? The option will obviously return either (0-premium) or (spot-strike-premium), depending on the strike. Your substitute hedge in the underlying will always return 0.
Imagine instead you buy a 100 ATMF call with a ~16% implied vol (ie ~1% a day), and the price goes 100, 101, 100, 101, 100... expiring at 100. Your hedge will "buy high and sell low" every day, losing you a lot more than the premium on the equivalent option you're trying to hedge.
The hedge guarantees the same risk exposure as the option at any point in time. This obviously prevents arbitrage between the option and the underlying. However, this does not then in turn guarantee the same payoff from the hedge as from the option.
Put differently, Black-Scholes is an unbiased and asymptotically consistent estimate of the value of an option... but the hedge that delivers these attractive properties is no guarantee in finite samples... ;-)