Israel has long prevented the import of all sorts of goods. They do not provide "everything necessary." Israel also has prevented the export of almost all goods, so that the Palestinians will have no money and be completely dependent on Israel to provide for basic needs.
I'm sure you're right that if the border were opened up that Hamas would import weapons and attack Israel, which is why Israel surrounded Gaza, as I said.
As you say, "peace requires proof", but that goes both ways, and ever since Camp David collapsed, Israel has likewise proved that it does not intend to make peace by further expanding the settlements, and generally making the lives of the Palestinians intolerable, often in capricious ways that do not enhance Israeli security.
The very obvious long-term goal of Israel is to drive the Palestinians out of the occupied territories and annex them for Israeli use.