Sie müssen die Diagnose schon Ihrem Doktor überlassen.

This sentence has one of those uses of “schon” that I hear Germans use frequently, but would never think to use of my own initiative. I also don’t think I understand fully what it means. Like what is the difference between this sentence with or without “schon” in it?

In a context like this it emphasises what you have to do in order to get something, usually as a way to tell you that you did something not the right way. Usually it reflects light annoyance.

Du musst schon drinken, wenn du nicht mehr durstig sein willst (dumb kid, did you think you could just stop drinking?)
Du musst schon auf das Ergebnis warten. (what do you think you’re going to get by pushing, just wait, geez)
Du musst schon auf deine Eltern hören (das hast du jetzt davon, silly boy)

Schon is very flexible in use so this limited to this context only.

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