Einstein's assumption that the speed of light is independent of the motion of the source was false but sounded reasonable - it is valid for all waves other than light so assuming that light makes no exception is justifiable. However, combined with the principle of relativity, the assumption entails an obvious idiocy - the speed of light is independent of the motion of the observer as well. Einstein saw the idiocy but didn't stop. If he had, physics would be still alive today:
John Stachel: "But here he ran into the most blatant-seeming contradiction, which I mentioned earlier when first discussing the two principles. As noted then, the Maxwell-Lorentz equations imply that there exists (at least) one inertial frame in which the speed of light is a constant regardless of the motion of the light source. Einstein's version of the relativity principle (minus the ether) requires that, if this is true for one inertial frame, it must be true for all inertial frames. But this seems to be nonsense. How can it happen that the speed of light relative to an observer cannot be increased or decreased if that observer moves towards or away from a light beam? Einstein states that he wrestled with this problem over a lengthy period of time, to the point of despair." https://history.aip.org/history/exhibits/einstein/essay-einstein-relativity.htm
The independence of the speed of light from the speed of the observer is not just nonsense - it is an obvious idiocy. Any correct interpretation of the Doppler effect shows this:
When the observer starts moving towards the light source with speed v, the frequency he measures shifts from f=c/L to f'=(c+v)/L=f(1+v/c), where L is the wavelength:
http://www.hep.man.ac.uk/u/roger/PHYS10302/lecture18.pdf "The Doppler effect - changes in frequencies when sources or observers are in motion - is familiar to anyone who has stood at the roadside and watched (and listened) to the cars go by. It applies to all types of wave, not just sound. [...] Moving Observer. Now suppose the source is fixed but the observer is moving towards the source, with speed v. In time t, ct/L waves pass a fixed point. A moving point adds another vt/L. So f'=(c+v)/L."
http://docplayer.net/35188128-Modern-physics-notes-spring-2007-paul-fendley-lecture-35.html "Now let's see what this does to the frequency of the light. We know that even without special relativity, observers moving at different velocities measure different frequencies. (This is the reason the pitch of an ambulance changes as it passes you it doesn't change if you're on the ambulance). This is called the Doppler shift, and for small relative velocity v it is easy to show that the frequency shifts from f to f(1+v/c) (it goes up heading toward you, down away from you). There are relativistic corrections, but these are negligible here."
Does this mean that the speed of the light relative to the observer shifts from c to c'=c+v? Yes. Consider the following setup:
A light source emits a series of pulses equally distanced from one another. A stationary observer (receiver) measures the speed of the pulses to be c and the frequency to be f=c/d, where d is the distance between the pulses:
where c' is the speed of the pulses as measured by the moving observer. Clearly,
c' = c + v.
That is, the speed of the pulses varies with the speed of the observer, in violation of Einstein's relativity:
http://physics.bu.edu/~redner/211-sp06/class19/class19_doppler.html "Let's say you, the observer, now move toward the source with velocity Vo. You encounter more waves per unit time than you did before. Relative to you, the waves travel at a higher speed: V'=V+Vo. The frequency of the waves you detect is higher, and is given by: f'=V'/L=(V+Vo)/L."
http://a-levelphysicstutor.com/wav-doppler.php "Vo is the velocity of an observer moving towards the source. This velocity is independent of the motion of the source. Hence, the velocity of waves relative to the observer is c + Vo. [...] The motion of an observer does not alter the wavelength. The increase in frequency is a result of the observer encountering more wavelengths in a given time."
http://www.einstein-online.info/spotlights/doppler Albert Einstein Institute: "The frequency of a wave-like signal - such as sound or light - depends on the movement of the sender and of the receiver. This is known as the Doppler effect. [...] Here is an animation of the receiver moving towards the source:
By observing the two indicator lights, you can see for yourself that, once more, there is a blue-shift - the pulse frequency measured at the receiver is somewhat higher than the frequency with which the pulses are sent out. This time, the distances between subsequent pulses are not affected, but still there is a frequency shift: As the receiver moves towards each pulse, the time until pulse and receiver meet up is shortened. In this particular animation, which has the receiver moving towards the source at one third the speed of the pulses themselves, four pulses are received in the time it takes the source to emit three pulses." [END OF QUOTATION]
"Four pulses are received in the time it takes the source to emit three pulses" means that the speed of the pulses relative to the moving receiver is greater than their speed relative to the source, in violation of Einstein's relativity.