Chris, the PvP section of this forum is mostly dedicated to people looking for their shardmates so they can coordinate attacks for the different payout times. People on the same server definitely currently have different payout times. If you are in a competitive arena you will intuitively see this by watching people's attacking behavior (each person will use all their battles in a certain hour each day).
Now if you let people choose their payout time, people may indeed be able to get an advantage by picking a less popular payout time (I imagine for time zones over the Atlantc and Pacific there aren't many players fighting for payout). But if everyone can do it (say one time only), it's not really unfair. Beyond the problem of people moving to a different time zone or downloading the game while on vacation, many have complained that their arena payout times overlap with dinner/family time. It might be a pain to code but would be a nice QOL improvement for many.