If anyone is interested... the condensed version of how this turned out:
I took the watch back to the watchmaker and asked him if he could improve the time keeping and beat error a little, and replace the broken screw. He said he could not fix the beat error because the movement was old (a different tune than the "it will work like it is new" that I was told originally) and could not replace the screw because they don't make those screws any more. I asked him why he did not mention the broken screw to me before. He told me he didn't have time to explain every detail to his customers and that the broken screw does not matter anyway, so he didn't feel the need to inform me about it.
He said a few other things that I found pretty disturbing for someone who is supposed to be a professional at this... but I don't think you all want to read a dissertation on the event.
In fairness, he also said that I could call back in a couple of months, and if he comes across one of the screws between now and then, he would hang onto it and install it for me. It took a while to get him to that point and I am not holding my breath for that outcome.
I guess the moral of the story is that for vintage Omega movement work in NYC, Manhattan Time Service is not the place to go, despite what their web page says.
My search for a good watchmaker in NYC continues!!! Suggestions are welcome!
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