I have a Father Time that had a split in the balance wheel and a broken staff. I have replaced the wheel after much swapping and poising to get it in balance with a new staff and it's ready to go. My question is, in the wheel there are four screws (12, 3, 6, 9, o'clock that have plugs in them to prevent removal. WHY?

That was Elgin's idea of "mean time screws". They studded the wheel at 4 places 90 deg apart with threaded shafts. Screwed on those shafts should be slotted cylindrical Gold "nuts" to provide "mean time" adjustment of the watch.

David, Thanks for taking time to educate me on that subject. I think I remember reading that somewhere but can't place the source. When I get a junk wheel I strip it down and keep any usable parts such as the screws.
This Father Time is whole again because of that practice.