I think you'll find many uni sailors in agreement that the main is what finally puts the boat over, even down wind.

Speaking for myself, when a hard puff hits under spin I naturally drive down but hold the spinnaker for as long as possible. That can be difficult as the sterns start to lift. As a last resort, once it's clear the boat won't recover I dump the spin and make a stab at releasing the main. When I'm successful, the boat recovers nicely then it's just a matter of reversing the sheeting order and off you go. The total time is rarely more than 3-5 seconds.

The point is: that flat head sail, high up is the culprit. Imo, uni sailors would be better with something more like a pin head.