My biggest fear would be that it would dramatically increase the white player's first turn advantage. At the end of the first placement, there would only be 160 possibilities. White has further chances to whittle down on his next placements, and you might be able to trim it enough to do some memorization and gain a huge advantage, exactly what 960 is supposed to avoid.
I don't know that, though. It just seems like it could happen.
For what it's worth, Burmese Chess starts by placing the pieces on the board, one at a time, subject to some constraints.
I just had a 'random' thought today: what if the piece positions for Fischer Random (960) chess weren't random at all, but rather were deliberately chosen by each side? To keep it balanced, the process could go something like:
This could add a new and interesting layer of strategy to the game could it not?