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This public fora myth seems to be spreading: batteries do not "drive AF". Camera and lens control circuits "drive AF".
Similarly, your 100 W stereo amplifier would not sound any louder/better if you plugged it into a European 220 Vac wall outlet.
Though without an adequate power supply, those control circuits will not be able to drive the AF as intended. Taking a different approach to your analogy, your 100W stereo amplifier will not put out 100W if it's plugged into an inadequate power supply.
In the case of the 1D-X, the camera drives the lens and the lens in turn drives the ultrasonic motors. Now there are two paths with which Canon could approach the AF drive:
a) Produce motors which require even less power for the same amount of operating speed and torque required.
b) Increase the output of the power supply feeding the motors and particularly the electrical current output performance of the battery (i.e. further decrease the internal resistance).
Unless any massive advancements have been made to the motors and/or the control circuitry, option b would be the more feasible one. Hence, the nominal voltage of 11.1V for the LP-E4N and the nominal capacity of 2450mAh. The additional power output from the battery allows for more power to make its way to the lens motors. It's no different to how a battery grip with dual batteries in other models marginally increases the shooting performance of those respective cameras - higher power output due to the ability to split the current delivered between the two batteries, thereby proportionally combating the internal resistance.
In summary, you can't have one without the other. I agree that the control circuitry is crucial to the AF and shooting rate performance but it's not achievable without a good power supply behind it.
Getting back on topic though, it's definitely going to be an exciting firmware release which I'll definitely be taking advantage of. Nice one Canon!