Somebody is the final legal decision maker. If there is information clearly not available to the captain and it would be unreasonable for captain to seek out the information under the circumstances, I think responsibility can lie elsewhere.
If captain should have reasonably sought applicable information re aircraft status, then responsibility is more obvious. The importance of the status would fine tune the eventual degree of captain responsibility.
Captain's first ETOPS flight should have had him on a heightened awareness. If the complete paper trail did not show ETOPS status for the particular aircraft for that flight, it would be very difficult to defend proceeding with the flight.
The decision making process is to justify proceeding, not to justify a decision to not proceed.