TAI provides a time coordinate generated by taking the weighted average of the proper times of 450 world lines tracked by atomic clocks. Like any other time coordinate, it provides a temporal orientation but no time coordinate could be described as "universal" or "linear" in general relativity. It would be a good approximation to proper time experienced by most terrestrial observers.
Note that general relativity doesn't add much over special relativity here (the different atomic clocks will have different velocities and accelerations due to altitude and so have relative differences in proper time along their world lines). If you already have a sufficiently general notion of spacetime coordinates, the additional curvature from general relativity over minkowski space is simply an additional effect changing the relation between the coordinate time and proper time.