"...is not wanting a game to "get away from you" in the event of a chain-autopass, such that when you look at it again you don't even recognize it as the same game you last moved on...."

I think it would be easily solved to still show every move for each play (with maybe an option to skip all autopass/move turns)

For for example.
Turn 1 - Player A is on line, makes a move in a game - submit. (Other player has an auto), game right away come back to Player A.
Turn 2 - Player A is on line, makes a move in a game - submit. (Other player has an auto), game right away come back to Player A.
Turn 3 - Player A is on line, makes a move in a game - submit. (Other player has an auto), game right away come back to Player A.
Turn 4 - Player A is on line, makes a move in a game - submit. (Other player has an auto), game right away come back to Player A.
Turn 5 - Player A is on line, makes a move in a game - submit. (Other player has an auto), game right away come back to Player A.
Turn 6 - Player A is on line, makes a move in a game - submit. (Other player now is able to make a turn, so game goes to Player B.

Later in the day, Player B comes on line - see's game, and SEE'S Turn 1 (Of course unless they have the skip option checked)
Turn 1 - no move, hits submit. Since player A already made a move, game comes right back to Player B
Turn 2 - no move, hits submit. Since player A already made a move, game comes right back to Player B
Turn 3 - no move, hits submit. Since player A already made a move, game comes right back to Player B
Turn 4 - no move, hits submit. Since player A already made a move, game comes right back to Player B
Turn 5 - no move, hits submit. Since player A already made a move, game comes right back to Player B
Turn 6 - Oh, they now are able to make a move - makes it, submits - and now game goes back to Player A

So 12 turns (6 for each player) is played quickly - moving the game along. Both players don't "miss" in action [unless then choose to skip it].

That way no one is "lost" or doesn't know what happens.