Why was the Special Theory of Relativity needed?

The velocity addition problem

The observed constancy of the speed at which light travels tells us that the Newtonian model of space and time is flawed. But the flaws don't become noticeable until we start trying to describe things moving near the speed of light.
One reason why Einstein's Special Theory of Relativity was needed was because of the special problems that cropped up when scientists noticed that the speed of light is a constant everywhere in every direction. This caused problems with the Newtonian model for measuring time. One of these problems is called the velocity addition problem.
The velocity addition problem is illustrated in the figures above and below. In the top figure we see the red driver's frame of reference or rest frame. To the driver of the red car in her rest frame, the blue car is travelling forward at velocity U and some purple object is flying out of the blue car at velocity V.
In the figure below the point of view of the blue driver is illustrated. To the driver of the blue car, the purple object is flying forwards out of her window at velocity V', and the red car is driving backwards.
The velocity addition problem asks the question:
Given U and V', what is V?
If we use Newton's model for time as being experienced exactly the same for all observers, we wind up with the answer: V = U + V'. How?
Suppose, as Newton believed, the red and blue drivers measure time and distance precisely the same. According to the blue driver, in time T the distance of the purple ball from the blue car is X_ball = V' T. In the same time T in the reference frame of the red car, the blue car has travelled the distance X_car = U T. According to the red driver, the total distance the purple ball travelled is the distance the blue car travelled from the red car plus the distance the purple ball travelled from the blue car, or X = X_car + X_ball = U T + V' T = (U + V') T.
But X = V T, and that gives us the velocity addition formula:

V = U + V'

So, for example, if the blue car is going 30 mph and the driver of the blue car measures the purple ball going 60 mph, then the driver of the red car should measure the purple ball going 90 mph, because U = 30 mph, V' = 60 mph and hence V = U + V' = 90 mph.
Sounds reasonable, right?
Okay, now suppose that U = half the speed of light, and V' = the speed of light (maybe the purple object is a laser pulse). Now what is V? The above formula tells us V = one and a half times the speed of light.
This is contrary to observed behavior of Nature. Therefore the Newtonian model of time as being experienced equally by all observers must not be a good model for Nature.