[Animats]

> Perhaps by using a sorter that puts the screws in a line, axially, without a preference for heads-first or threads-first orientation?

Here's a vibratory bowl feeder doing exactly that.[1] This is the industry standard way to solve this problem. Look what happens once the screws are lined up without a preference for heads-first or threads-first. A very simple slotted rack gets them all from horizontal to heads-up. As is usual with such feeders, if something doesn't land where it's supposed to, it falls back into the bowl for another try. That's the anti-jam mechanism.

3D printing vibratory bowl feeders works.[2] Useful for when you need to handle thousands, but not millions.

This is more scale than the clockmaker needs, though. Unless his business scales up.

[1] https://www.youtube.com/watch?v=kIPNdrbSYM4

[2] https://www.youtube.com/watch?v=ECx6L7z0T4Y