I found it!

I thought it was gone, but I stumbled across it when cleaning. What is it?

I managed to get out the part that snapped and now I understand why it broke (besides my bumping into it): the threaded portion is pretty small in comparison to the diameter of that shaft. There’sa lot of tension in that center post, so no easy way to reattach it.  I guess I’ll have to buy another whole assembly instead of repairing this one. Chances are that this unit would sell for more if it was intact rather than having the next owner replace that part, just like selling a used car - fix it before the sale and get a higher return.