There is generally a slight bias for each cell towards 0 or 1 due to process variation, so when you completely kill power and restore it the cell is more likely to come up in one value than the other

Then the right way to interpret BER=0.5 is: half the time the bit powered up in an advantageous way, and the other half it didn’t? And a BER=0 implies no bit needed to turn on.

Yes, exactly. Put another way, BER=0.5 means the data was completely randomized