Historically nations only commissioned privateers in times of war, and the privateers were only empowered to attack nations at war with whoever issued their letter of marque. So the game would actually be far LESS historically accurate if you could do this “outside of declaring full blown war”.
Privateers were not pirates. Privateers were legally recognised legitimate combatants in a war. Pirates were criminals, reviled by all civilised nations, and deemed enemies to all humanity. In times of peace, former privateers who suddenly found themselves unemployed sometimes had little choice but to turn to piracy, but this made them criminals subject to summary execution if captured by any nation, including their former employers.
To get the effect you want and still maintain any degree of historical accuracy the game needs different types of war. In all Civ games that I’ve played, being at war means being in a state of total war as we’ve come to recognise in the 20th century. But in earlier ages in real life being at war didn’t necessarily mean gearing your entire economy for war and marching all the forces you could muster at your enemy’s capital. Being at war could be more of a diplomatic and symbolic thing with little to no actual combat involved, or it could mean border skirmishes and colony raiding.
The problem with warfare in Civ is that we only have “full blown war” as an option. They should add an option for colonial warfare where you cannot enter enemy homeland territory but can freely fight in neutral territory (like the sea) or raid / capture distant colonies. This type of war should have no war weariness (or very little) and have only minor diplomatic penalties compared to total war.