Ok, so I found
this video of a Singer 237, and it has a similar mechanism for the zigzag. After watching that, I believe that my machine should move in a similar way. In order for that to happen I think that it must be missing a gear/should be engaged with the drive shaft in this area, which would rotate a cam under the sliding piece allowing it to move forwards and backwards (zigzag). In other words, the entire slider should pivot. Basically the part I've drawn an arrow to should probably connect to those teeth right beside it, but there's a gap here, and clearly that isn't the right type of part that would connect anyway.
