I am trying to understand how the NAND-Flash memory works and I think I understand how programming a page works but I don't see why inverting all the voltage would not lead to erasing the page. So, why is it that the whole block has to be erased (instead of a finer grained page-erase)?
What you say is more similar to old EEPROM and NOR Flash, in which the erasing process basically consists in the injection of the same amount of opposite charge. The problem is that the cancellation has to be really accurate, otherwise the risk is to not discharge completely the gate or charging it with an opposite value.
The Flash NAND memories use the tunnel release principle, that (if I remember well) basically consists in making the charge flowing away from the floating gate to one of the active regions, avoiding the risk of injecting other charges.
Probably it's not so clear, but the reason why it doesn't discharge with an opposite voltage is because with NAND Flash you discharge the Floating Gate as you would do with a capacitor, simply letting the charge flow away; while the other technologies require the injection of an opposite charge.
(Take this answer with caution, it's not fully accurate but based on what I remember from the course...when I'll get the material again I'll give you a more accurate answer)