No, you will not need the Flintlock Pistol. You can progress through the story generally without needing to acquire the Flintlock Pistol. The Flintlock Pistol is only a cool extra for players to discover. Unlocking it will require players to explore more thoroughly. It's a very powerful weapon that may seem like an easy mode for players, so unlocking it is completely optional and can be used to beat some of the more tough caves with ease.