In General Relativity, defining the fourth coordinate as time is necessary to establish similarity with the special relativity. In General Relativity, there are no "bilocal vectors" (See the book by Misner C. W, Thorne K. S., Wheeler J. A.: Gravitation) and therefore it is not possible to define the velocity or force vectors. But, the special relativity deals with velocity and force vectors. Therefore, the space-time is supposed to be globally curved and locally flat so that principles of special relativity can be locally applicable.
See the preprint on ResearchGate:
Vector analysis in the curved space-time: The problems, the remedies and some concerns
The classical vector analysis ("bilocal vectors" method) method gives all the desired results the same as tensor analysis dealing with only the relations between scalar components of a tensor. But, this vector analysis also raises some serious geometrical concerns about the curved space-time. We see that locally the curved space can be only called as approximately flat which is not same as flat. Hence, it will be difficult to define vectors, such as force, even locally.