The conception of time in special relativity has much been influenced by the classical idea of absolute time. Absolute time can be represented by a geometrical straight line T. Points on T correspond to moments, while distances to durations. Every event is mapped to exactly one point on T. In other words, there exist absolute temporal relationships among events.
Looking back, the tacit motivation has always been to gradually reintroduce absolute time into the new theory, by providing feasible ways of measuring it in increasingly general situations:
Local time: multiple observers at the same location in an inertial frame all sense the same objective time passing that can be measured by a co-located (local) clock.
At that point, the possibility was open that all local clocks in fact measure absolute time, identifying points and distances on T.
Inertial frame time: the expectation that objective simultaneity must then exist within a single inertial frame led to a well-defined coordinate time, measured by the synchronized local clocks in the inertial frame.
At that point, the possibility was open that coordinate time in fact (necessarily) measures absolute time on T as well.
After that, our unspoken attempt came to a dead-end when multiple inertial frames were considered: it was demonstrated via the time dilation effect that coordinate time cannot measure absolute time, since both simultaneity and duration are inertial frame dependent. To be precise, from the perspective of absolute time what was demonstrated is this: when two inertial frames are moving relative to each other, it cannot be that the coordinate time in both of them measure absolute time. Yet the principle of relativity suggests that the situation of the two inertial frames must be symmetrical. It would not fit in the picture if it was possible that the coordinate time in one frame is absolute, while in the other it's not… and to make matters worse, there would be no known way to tell which frame is which.
This is the point where we must realize that time dilation has left us with no tangible evidence but only one thing supporting the idea of absolute time: our imagination, i.e. the intuition that we've gained from the limited spectrum of our day-to-day experiences. Time dilation has refuted the strongest argument we thought we had for absolute time. Namely, it's not true that two clocks of the same construction always show the same time elapsed between any two of their encounters. To test this, we don't even need to define simultaneity and coordinate time, all we need is the two clocks.
That's why, since it does not seem to help to go back and try to adjust the simple and robust definitions that led us here, the choice has been made in special relativity to rather give up absolute time.