when an inspector is using the survey on site i want the initial "who is the surveyor" question to become a default for the rest of the session?
By "become a default for the rest of the session" do you mean for all new surveys collected after the first one? Note that the calculations, defaults and other settings configured in the survey only apply to that survey record (feature) and its related records and are not shared across different surveys (features) when opening a new collect survey entry.
However, have you considered using the signed in user to set the default you want, this can be done with a calculation in the survey for instance.
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