It would be really great if you can build and live on the deserted island using no money. So it is an actual survival experience. In another topic someone was suggesting something similar about gathering sticks and stones to craft items and I want to expanded on that idea. I really like your idea about farming plots being there to make your own food and the fruit trees on the island to gather foods. For this idea to let it actual work better it would be indeed neccesary for the island to become bigger so there are more resources to gather.
Instead of money your zoi gathers resources, like sticks and stones to craft items, furniture and a shelter. Examples of collectables would be, sticks, stones, wooden planks, nails, leaves, rope, plastic, fabric etc. There should be different ways to gather these materials. You can find materials on the ground just like sticks are currently in the game. They could make it that your zoi can go fishing and sometimes find usable collectables instead of fish. Or when opening treasure chests they sometimes include these resources. To expand this feature to outside the island, they could also think about adding a functionality to search the trash bins to find usable resources as well. Maybe they can find old trashed furniture in the trash that they first need to repair with planks, sticks etc. before being able to use them.
It could be great if they would add a usable hammer that can be crafted on the workbench just like the pickaxe and fishing rod. Your zoi needs to have a hammer in their inventory to be able to repair and build objects and there is a chance that the zoi does not succeed and use up all the ingredients for nothing, the chance of this happening is based on their skill level. They can destroy existing (trashed) furniture with the hammer to gather some resources. For example, destroy an old table to gather 2 wooden planks and 1 nail. So the hammer could be used to repair, destroy and build.
They could extend it with a craftable axe (again on the workbench), which can be used to cut trees for wood planks, sticks, fruit and leaves. And also chop down bushes to gather materials for rope. And the use for the pickaxe could be expanded to be able to harvest stone and gemstones on rocks that spawn on the island.
With the new update, when diving you can find some collectables as old furniture or planks etc on the ocean floor. And sometimes collectables wash ashore which respawns in a couple of days. If you are lucky even a golden chest washed ashore which contains some special items, like equipments (watering can), food (canned beans), seeds, furniture, decorations, blueprints etc. So when walking on the beach you can sometimes find resources to collect.
I was thinking about a seperate survival skill which unlocks blueprints to build items. Some blueprints unlock while leveling up the survival skill, but some blueprints can only be found in chests (but the ability to be able to build these items need to be tied to the survival skill as well). These blueprints are the items you learn to build with the hammer. For example, you start with sleeping on a bed made with leaves or the cardboard bed and a basic fire. And when leveling up you learn blueprints to build more advantaged items as stone walls, a normal bed, fire place, water collector.
This would really give a rags to riches, stranded gameplay a next level experience. It would be great if your zoi could use this method to eventually build up their whole house 
) it will be more filled in. I hope this applies to many areas — the second floor of the beachside café is still tempting with its closedness, and I really want to wait for when it finally opens 