How to Extract Broken Left-Hand Screw?

I ordered a new drill transmission for $64. I can't order the spindle by itself, so I bought the whole transmission.

I took the drill apart to see if I could find a way to remove the spindle and work on it from the back end. Lost a couple of detent ball bearings I didn't know were there. They took off. No idea where they landed. I realized I was now looking at waiting for the extractor AND measuring the remaining ball bearing and ordering two new ones. And I was probably going to be unable to remove the screw when everything was put back together.

I had already ordered two replacement screws, so I am ready to attach the chuck when the transmission arrives, over two weeks from now. I have one screw to make the drill work, and one in case I ever break the new one.

I think I should put anti-seize on the screw threads. Conceivably, it would prevent the new screw from becoming permanently stuck if I break it off.
 
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