It might be a bit confusing at first (due to the misnaming of the method), but technically the pico does not use deep-sleep, but powers down. A button-press or an interrupt from the rtc enables power again, and the pico boots and your program runs again from the beginning.
When on USB, the pico is not powered down but running idle and your while-loop is running. Otherwise the while-loop is only executed once.
When on battery, I suppose you have tried to wake up the inkyframe once using a button (necessary) and then waited for the next startup?