You can beat it, but you'll be missing a good chunk of the story. Basically, if you don't do the entire set of sidequests with Keith on the island there you're going to be missing out on "hows," "whys," and "whats."
Said hows, whys, and whats are also relevant to a main conflict in Escha and Logy.
Belated Edit: Anyway yes, if you just want to beat the game you only need to get a certain number of flowers, then an event battle with a mean machine opens up that will probably kill you a few times until you make better gear and items. This is the trigger for the rest of the events in the story. More things will happen and eventually a final dungeon opens up. You'll know it because it's gorgeous.
But the final boss won't make a lot of sense unless you did the Kieth events.