In 1916 Einstein published an argument to show that the relativity of simultaneity [RS] could be deduced without knowing more about special relativity [STR] than the constancy and invariance of the velocity of light. (Relativity: The Special and the General Theory. 1st English tr. 1920, pp. 27-29.) I believe the argument can be improved.
The idea is that a moving train is hit by lightning at both ends. An observer on an embankment is at the mid-point of the train at the instant of the simultaneous lightning strikes. Hence this observer sees the flashes of lightning simultaneously. A passenger in the train is sitting at its mid-point. Because the train is moving forward, the passenger sees the front strike before the rear strike, but this is in the frame of the embankment. Einstein claimed that for the same reason, the passenger in the train would also see the front flash first, thus illustrating the relativity of simultaneity. The logic of this seems unclear. Without a transformation formula, we have no way of knowing what the time coordinates in the moving frame are.
One can argue like this: Either the times of seeing the two strikes are the same in the passenger’s frame or they are not. If the strikes appear simultaneous, this is RS for the events of seeing the flashes. If the strikes appear non-simultaneous, the strikes happen at different times in the passenger’s frame, because the flashes travel equal distances to the observer and at the same speed. This is RS for the events of the two strikes.