As a sitter and owner with many sits and sitter experiences I can honestly say when on a sit I have never opened a draw, cupboard etc that I have not had reason to. I am not remotely interested in anything outside of the pets in my care and the parts of the home I need to use. I will run a vac and duster around rooms I don’t use but otherwise my focus is solely on my responsibilities and I truly believe that is our way, that is why our community works.
As an owner I have treated my sitters in exactly the same way but we each have different comfort levels and owners need to do whatever is right for them, including what to do with precious items.
A locked room is certainly not going to offend me but as @Snowbird says make certain that it is a room where access is not needed for any essential maintenance.