I'm talking about the old-school theodolites that surveyors would have used in the 18th and 19th centuries. Could people in those days have navigated using such instruments? If so, wouldn't they only have worked if the user knew the coordinates of the landmark that he or she was taking the sighting on?

Much obliged.

A theodolite is basically a more accurate and sophisticated sextant. People have used them for navigation for centuries.

You don't need to know absolute coordinates. Relative angles are enough.