SpringHeeledJack,
I don't remember where I read it, but the advantage of not allocating seats is a faster boarding. People tend to line up at the departure gate much earlier and take the first free seat they like hence the whole boarding procedure is much faster. In an assign seat system people know they have the seat they like so they take their time to reach the gate, go missing...
In the proposed system, I don't understand how is it going to work if a passenger who has not paid for a specific seat takes his/her chance with the current rush to get a seat only to find it has been assigned by the check in to another passenger.