I'm currently on even goal 6/40 "open an event chest once." However I've already cleared the first two rooms and can't unlock the 3rd room until I complete 8 event quests or pay $20. Not sure if I'm missing something, but seems like I'm unable to progress at all in the event now. I've tried refreshing the game but doesn't seem to help.

You get a trivial amount of silver from tower chests.  I think the intent is to grind away at those for a couple days and act as a time sink against free players.

You can get up to 75+100+125+150+175+200+(50+75+100+125)+(150+175+200+225)=825+350+750=1925 Pirate silver.
So if you don't have a event chest key, you have to get 75 silver in tower chests(a big rare chest, or a rare chest & a common chest, or 3 common chests) to complete 'Open an event chest once.'