As it turns out, Pirates didn’t drink “barrels of rum”, they drank wines, ports and other stable liquids. However, the Navy did have a hand at rum based drinks like Grog. The Pirates, they had products like Madeira which was used to create some cocktails.
This Madeira made it to some of the noble houses, some of the larger high class parties in the form of juice-based drinks. Not quite a punch, but pretty close. Those higher class folks had access to cognacs, sugars and spices like nutmeg so they built stuff like the Sangaree.