> Did that work in the 18th century?

It did in the early 19th century. Check out the first and second Barbary Wars. They were not permanent solutions but they had lasting effects. The real blow was the French conquest of Algeria after that.

1. Doesn't seem like this quite fits under "hanging a few pirates"

2. One of the big problems here is there's no real counterparty to negotiate with/cow like the Algerian Dey.

That's not to say that I'm against increased anti-pirate action. I just don't think it's as straightforward as you all say.