First off, I have never seen an A16 magazine. I have used and repaired (professionally) many A12 magazines of various vintages. The two are almost identical.

I checked the service manual, which covers all of the magazines. There is no mention of a "stick" in the exploded parts diagram nor in the parts list.
There's also no reference to such an item in the user's manual. I find no such in any of my A12 magazines that I opened up to check.

It sounds to me like there's a problem in the film counter, and somebody provided this "stick" as a way of resetting the counter, thus allowing the back to function sort of.

- Leigh