Technically the only thing you need for boss access are the key items from each branch's boss. In order to get those, you need all expeditions on that branch completed to at least goal 1. Higher goals give you better boons which are no longer needed with item level gear. They also give you access to buy certain pieces with marks, which are usually junk now.
You can also buy the key items with gil after winning them at least once. 1.2 mil per pop set for Batallia, and 1 mil per pop set for Sauromugue