Less volume with a more powerful amp

Amp gain (sensitivity), not output power, determines the necessary level setting.

Is particular amp sensitivity a predetermined engineering goal for a given design or is it an incidental outcome?

I do not think there is a single answer. Since it is interdependent with other design parameters, it is not an independent variable.

Obviously, sensitivity needs to fall within a useful range for commercial success but some designers have specific targets and others do not.

I don't mean to pester you but is high sensitivity commensurate with a low noise floor given all other performance parameters are of a high standard?