As I understand it, the reason why the recorders aren't usually powered during time on the ground, is to preserve the recorded data. Power for the recorders still comes from the essential busses as required by the design regulations.
Data recorded all the time the aircraft is powered up would overwrite flight data with useless ground data. A typical airliner is powered up pretty much constantly between the power-off stages of major checks, but will be airborne only about 60% of the time - 14 flight hours / day would be considered a typical utilisation for a civil airliner. (The highest utilisation I can recall from my own civil operations experience is 16 hours/day.). So a permanently powered FDR would have about 40% of its useful data overwritten by irrelevant ground data. As I pointed out above, we "back room" engineers and Flight Ops QA use DFDR data for much more than accident investigation.