Replacing setting lever screw on a cal.1570 movement

[Fioran0]

Apologies for cross posting. Stuck this question into the vintage section. But too late realised this was probably a better home for it. MODs feel free to delete the other if cluttering things up.

Hoping anyone who was delved into the vintage mechanisms might be able to help with info on replacing the setting lever screw (stem retaining screw) on a cal 1570 movement.

Watch started misbehaving over the holidays. Crown would need an additional pull/push to get going after setting hands. Escalating firstly to extra finessing and then finally crown/stem just coming out all together rather than setting anything.

Opened up to find stem screw loose and spinning but unwilling to tighten. Removed dial and found setting lever itself loose too. Small bit of crumbly material in the setting lever which cleaned out ok to leave the appropriate hole. But no section of the screw thread underneath protruding to reattach. Assuming its either sheared, or more possibly been short for a while and ive simply been riding my luck.

Wanted to ask a couple of questions. First to check sanity. Second a whats next.

Am i right in thinking that the stem securing screw/setting lever screw is the same thing. The screw inserts through the movement from the auto winding side, threading into the setting lever from below to hold in place. Tightening the screw pulls the setting lever against the stem to lock. Loosening does the opposite. Seems logical but unable to visually confirm.

If above correct. How does one replace the screw. Im 50/50 on whether its captive underneath the automatic winding bridge - or indeed the plate below that on this movement. Or whether it should just come out as is by giving it a push.
Would rather ask before going exploring. Especially since replacing screw/lever from current situation looks ok. Stripping things further something Im less keen to try.