Zoom/Rush poker type of player batch seating distribution : not even

Question

Assume that you are developing player batch management for Zoom/Rush poker type of game.

Conditions:

1. Once a player folds, he moves to free player batch.
2. You have to place him in optimal place in pool for next game
3. Assume that current folded player is BB
4. Assume that 48 players are in pool (waiting for next game to start) for 6-max seater table

Placement criteria in batch

1. Take 6 player batch 8 times for 48 players
2. Sort the list with players in descending order of number of games played since last big blind
3. Assign BB to the first player in the sorted list
4. Assign SB to second player in the list
5. Assign Button to third player
6. Assign EP , MP and Cut-off
7. In case of players having same hand count, assign the position based on time at which player joined the free pool i.e First come, first served

Now when would these numbers converge to have even distribution for all positions.

If you play 6000 games, what is the possibility of getting each seating position uniformly 1000 times ( 6 positions * 1000)

As per my observations, even distribution for position is not happening at any point of times. Dealer/SB/BB or more less evenly distributed. But to EP/MP and CO.

Some times, the batch size is 4 or 5 depending on player count during that time. In these cases, I am considering seat 4 and 5 as CO instead of EP/MP.

Are below seasons causing deviation for even distribution?

1. Player playing on multiple entries
2. When the player moved to a batch, he competes with 6 players in his batch
3. Always you can't have 6 players in batch. It may be 3,4,5 too

If you consider above events happening randomly across multiple tables and entries, how should we guaranty even distribution?

Is there flaw in above batching logic?