This is a GREAT start, I love almost all of it 
That said, here I ask a few questions (and make an attempt at answering them)
- Why are those canes there? If they are so high that the player cannot pass through them, then no one else can too, right? So why did nobody did cut them already? As it is now, the BH is the only one allowed to get to that part of the island (using his own house). And since that is the most accessible part of the island (for you can dock boats there) I’d consider this a little weird;
- How exactly does the tutorial happen? Is there a textbox telling us what to do, with words spoken by no one but the game itself? Or is there a character speaking to us? And what event makes the dialogue box appear? Us getting close to the BH house? There is a sign with written instructions that we have to interact with to read?
- How do we know that there is a cooking festival going on? Does the BH know about it? And what is he doing under that tree?
There may be many answers to these questions, but here is a solution that tries to answer all of them at once:
My first drawing represents whan the player sees when the cutscene finishes and Hamlet steps on the island: a wide beach with some trees, and a more detailed point of interest (very noticeable also thanks to the rule of thirds) in the distance.
The player can explore this area, but there’s not much he can do (if not maybe collect some random low-value ingredient from the ground) and after learning the movement controls he heads to the point of interest.
Here I merged down the BH’s house and @itsLevi0sa 's idea of the bridge between the beach and the rest of the island. BH’s house is build (or better, digged) on two levels, with two doors. The great obstacle this times aren’t wild canes, but some crates and chests that someone put there. There a sign on one of them. It reads “Bard Hare’s food supplies. DO NOT TOUCH, STEAL OR INTERACT WITH IT IN ANY WAY”. Immediately after the player stops reading, a popup appears, in the middle of the screen, saying “Press X to destroy the crates” (I played it in my mind and tbh it could be hilarious).
When the player destroys the first crate, we hear a scream from somewhere. I don’t have an exact line in mind, but something like “Who is stealing my stuff?”
Camera shifts and we see the BH, looking at us. When he comes close, he doesn’t recognize us, so he guesses we’re foreigners. He realizes that with all his stuff obstructing the road he wasn’t allowing anyone to pass, and he apologizes, adding that he has too many things going on on his mind. He even alludes to some kind of event that could make him famous in the whole island, but he doesn’t elaborate on that.
He says that if we want further details on what he was talking about, we must do something for him. Somewhere around the beach there should be a “rare” ingredient that he can’t find, and he needs it for a special recipe. We have to get that ingredient and bring it to him. We can take all the time we want: it will take a while to remove all the crates from the passage, the’re pretty heavy…
Here we can explore the the island again, this time with a bit more freedom: we can destroy stuff, and even burn canes with our chick (maybe the BH tould us how to do that), and behind some canes is exactly where we find the thing that BH was looking for.
When we bring the item to BH, we found out that he has removed all the crates (until now he was standing in front of them, blocking us from destroying them) and the road is clear. As a reward for the ingredient, BH tells us about the cooking competition and wishes us good luck in case we want to participate.
Edit: Ok maybe the bard hare is’t competing in a cooking festival (or is he? Who said we can’t have more than one passion?), but this could still work. Maybe he was asked to write a song for the cooking festival, or one dedicated to the Elder Chef, or to do something else. All details here can be added/removed/changed 
