grab 4+ coins quickly in the early game and have a way to likely be played 2+ times per turn in mid game.
Just want to point out that the seventh shuffle (the bare minimum four attacks + 2 payload plays could take with one PS) is very rarely the "mid game"; it is at best what, Turn 15? You've already lost by then.
PS is just too slow to work without +Buy and some other method of lengthening the game, and it is just not good enough in 2P. Miser is literally a *pirate ship that attacks yourself* and it's widely considered to be the better card.
In my earlier posts, I've emphasized a way to play Pirate Ship multiple times per turn, such as combining with a strong village, which does much better than above in kingdoms where you'd want to play PS. To confirm, I wrote a sim for this case. The sim found that Pirate Ship + Fishing Village vs a decent strategy that emphasizes money typically reached 4+ coins around turn 8. Sims had an average game length of ~16 turns (assuming no opponent attacks or trashing), with Pirate Ship usually buying 4+ Provinces by the turn 15 "mid game" you listed. Pirate Ship soundly won in all such simulations of kingdoms/opponents that meet these criteria.
The game length increased in other kingdoms with opponent attacks, trashing, or emphasizing VC. For example, Pirate Ship + Fishing Village vs Smithy/Salvager + Fishing Village was an interesting case. If the Smithy/Salvager plays with a strategy that emphasizes virtual coins by only purchasing 1 Gold in treasure (no silvers unless FV runs out) and waits for $13 in deck for the first buy, then the game lasts an average ~20 turns and Pirate Ship had a 71% win rate. However, if the Smithy/Salvager player tries to speed up the game by buying provinces any time he has 8+ coins, then the game lasted an average of ~17 turns and Smithy/Salvager had a 54% win rate. If Pirate Ship modifies usual strategy to compensate for the modified fast game opponent strategy, then Pirate Ship wins the majority again. Smithy/Salvager could further modify... and so on... it can lead to some interesting strategy decisions. The games where Pirate Ship + strong village failed in my sims are the ones where you'd expect, which usually involved kingdoms with some combination of the following:
1. A strong attack, particularly cursing or hand size
2. A strong source of at least +2 virtual coins (played multiple times, with the village)
3. Possibility of a draw full deck type engine that includes multiple buys
4. Fast early game treasure trashing