For the benefit of anyone else who runs into this:
To get this working, I had to push the @State var playMode up to the parent view and pass it into this view as a binding.
I then had the parent view detect when a single Bin was selected and use a custom get/set binding to access its value directly rather than using the @State var (but still used the @State var when multiple Bin objects were in the selection list), and upon change to the playMode, had the onChange(of: playMode) toggle a boolean in an environment object in order to cause the view hierarchy to refresh.
I was then able to eliminate the getPlayMode function completely.
This seems to have given me the behavior I wanted, though it does seem a bit convoluted.
SwiftUI as a framework clearly has much room for improvement.
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It looks like the forum messed up the formatting of my list of instructions; adding bullets to try this again:
set
set 0
set x