I do not see how your analogy wrt lines of latitude shows the existence of spacetime. Analogies, metaphors, etc., are handy to convey principles, but if the principle itself is not valid, then the analogy is meaningless. Spacetime is a mental construct used in some models of the universe, but there is no proof that spacetime exists as something real as space is real, or matter is real.
Space and time are equally real. Motion can only happen because there is time and space together. But time (and space) are within the universe. The universe itself is through all space and all time. Hence it has a spacetime geometry. That geometry can be curved like the surface of the Earth (time corresponding to latitude in this analogy).If it is curved enough, it can even be finite.
The analogy is valid, at least as a possibility. Latitude and longitude are also 'mental constructs'. Neither is dictated by the geometry of the Earth's surface. Yet they are also quite helpful for describing where things are.