1) The electronic KERS used by most (all?) the teams is essentially an electric motor/generator attached to the front of the crank with capacitors for energy storage. Under braking, the motor is driven and charges the capacitors. When the KERS button is pressed, the motor drives the crank directly, essentially increasing the power output of the engine.
If the teams aren't doing this during overrun (i.e. whilst braking), they must be doing it during acceleration, which logically gives no benefit at all since power will be used to charge the system (almost the exact opposite of what they are trying to achieve).
2) The extra quantities of fuel was in relation to your suggestion that the teams are arriving into corners with the car in neutral and the engine revving (to drive the diffuser and recharge the KERS). This is simply not the case.
3) It's known as "hot blown" only in the media. Diffusers have been driven by the exhausts for as long as I can remember. It was only during the 2000's that exhausts were exiting from the sidepods, to drive the rear wing as I recall.
4) The ignited fuel into the exhaust is not "extra". The teams are maintaining 100% throttle during braking. Obviously, you don't want the engine to continue producing power (i.e. trying to speed you up) because you're trying to slow down. To counter this the teams don't ignite the fuel (the ignition is retarded to achieve this). You now have an engine that is maintaining maximum airflow through the exhaust but producing little power.
5) I don't see the relevance of your comment. The teams wouldn't happily forego either. They would look at the laptime benefit of either versus the weight penalty of carrying extra fuel.
Finally, what was the FIA's proposed method for stopping teams doing this? I quote, "the FIA has deemed that throttle use will be allowed only for the purpose of increasing torque, not for 'aerodynamic performance'." You see, that's where the movable aerodynamic device bit comes in. They're using the throttle (movable) to control flow to the rear diffuser (aerodynamic).
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