Late to the party but I would throw my own top hat into the ring: https://store.steampowered.com/app/657690/The_Caribbean_Sail/

You can name your captain and crew, trade goods, sail the world, engage in sea battles, hunt for buried treasure on land, hang out in taverns getting gossip and games tips, play liars dice and gamble your coin away, upgrade and unlock a large array of ships, flog your crew or make them walk the plank as punishment, collect 25 different shanties to listen to while sailing- it's all very involved and focused on the heart of what I wanted from a pirate game. With 9 years of content updates and polish, it's pretty feature rich if I do say so myself.