While 16 is fixed, the other 15 players do move around as evenly as possible. This may be an advantage in some situations, for example if the event organiser is participating and it is more important that everyone knows where to find them.
I am not aware of any schedules where the 16 players are evenly balanced across the tables. I have tried to improve the standard schedule by moving games between tables, for example I get:
[(A B):(G L)] [(C D):(F J)] [(H K):(O I)] [(M E):(N P)]
[(D E):(G K)] [(A C):(H M)] [(I L):(P J)] [(N F):(O B)]
[(E F):(H L)] [(A D):(I N)] [(J M):(B K)] [(O G):(P C)]
[(P H):(B D)] [(F G):(I M)] [(K N):(C L)] [(A E):(J O)]
[(A F):(K P)] [(G H):(J N)] [(L O):(D M)] [(B I):(C E)]
[(M P):(E N)] [(H I):(K O)] [(A G):(L B)] [(C J):(D F)]
[(N B):(F O)] [(I J):(L P)] [(A H):(M C)] [(D K):(E G)]
[(O C):(G P)] [(J K):(M B)] [(A I):(N D)] [(E L):(F H)]
[(K L):(N C)] [(A J):(O E)] [(P D):(H B)] [(F M):(G I)]
[(G N):(H J)] [(L M):(O D)] [(B E):(I C)] [(A K):(P F)]
[(H O):(I K)] [(M N):(P E)] [(C F):(J D)] [(A L):(B G)]
[(A M):(C H)] [(N O):(B F)] [(D G):(K E)] [(I P):(J L)]
[(A N):(D I)] [(O P):(C G)] [(E H):(L F)] [(J B):(K M)]
[(A O):(E J)] [(P B):(D H)] [(F I):(M G)] [(K C):(L N)]
[(B C):(E I)] [(A P):(F K)] [(G J):(N H)] [(L D):(M O)]
where most players are assigned between 2 and 4 times to each table.
Hope that helps.