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Think about the physics. In a typical flash circuit, the fully charged capacitor is floating at ~300 V on its terminals. When triggered, it is connected to a step-up transformer moving the ~300 V on the capacitor terminals to a few kV, enough to overcome the ionization potential of the xenon gas in the flash tube and cause an arc. This produces the flash seen.
The capacitor provides a fixed impedance to the circuit, so a high voltage pack would indeed stuff charge into the capacitor fast (i.e. shorter time constant). But all this does is get one faster to a fully charged capacitor. This is why a set of four AA batteries (nominal voltage = 6V) can, if given enough time, supply enough charge to the capacitor to prime it for a flash.
So in other words, the voltage of the pack is a red herring here. As long as the pack has enough CHARGE to fully load the capacitor, this is all that matters.
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