There was definitely a bug in the schematic in the opening post; the reset pins of the NE556 should have been connected to 5V an not to the trigger pins.
I think that I've found a purpose for the two remaining gates of the 74HC05. In the below design they will (or are supposed to) prevent both FET outputs being active at the same time; this will make it a little more idiot-proof.
I think it's time to buy the components and start building a prototype.
Updated schematic:
