"...but without that screwdriver slot I'd have never gotten the bobbin winder off of the machine."
It sounds like you've answered your own question here - these machines were designed to be easy to maintain, they had to be, for use in an industrial setting.
When you this part was assembled, how would you align the two holes for the set screw without something to mark the rotational position of the post, and something to turn it accurately? The screwdriver slot has both functions.
I started writing this, as a solution, but I didn't end up on a neat solution (although I am certain it exists). In any case you can find a solution with numerical methods at end. If I find the neater way I will update.
Assuming that :
the end of bar s is point S
the end of bar p is point P
the length of bar p,q,s are respectively $L_p, L_q, L_s$