Experimental proof of light speed isotropy Where is the experimental proof that observers travelling toward or away from a light source, will always find that light from that source measures the same speed, regardless of their own speed?
 A: There have been very many tests of the constancy of the speed of light. You can readily find lists and descriptions of them on the Internet. For example, see https://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html
A: It is claimed that on the earths surface the existence of an Äther would be observed with the Michelson-Morley experimen.

Michelson expected that the Earth's motion would produce a fringe shift equal to 0.04 fringes—that is, of the separation between areas of the same intensity. He did not observe the expected shift; the greatest average deviation that he measured (in the northwest direction) was only 0.018 fringes; most of his measurements were much less. (Wikipedia)

By this the quite possible assumption that the velocity of EM radiation depends from some property or conditions of space (Äther) was declined. Instead it was claimed that the speed of light is independent from the observers velocity.
Later Einstein introduced an observer in a point of space with different gravitational potential should observe a smaller value of c in a point of space with higher gravitational potential [like for example near a Black hole). The Äther was banned but isn’t the gravitational potential a condition or property of space?

Where is the experimental proof that observers travelling toward or away from a light source, will always find that light from that source measures the same speed, regardless of their own speed?

Reproducing the MM experiment on a satellite with its vanishing gravitation in relation to the earths field, with its high velocity in relation to the earth and it acceleration-free behavior your question will be answered better than with MM experiments on earth. But this experiment is not necessary because we are sure to reach the right answer for all time.
The point is that in our daily life we could live with the constancy of light independent from the observers speed or with the constancy of light due to the gravitational potential. More interesting is the question, would this two different points of view change something in calculations for real physical processes, say for particle accelerators.
A: First, the constancy of the speed of light and its isotropy in a vacuum does indeed have strong theoretic descriptions and proofs, such as those provided by Albert Einstein and Hendrik Lorentz.
Our everyday experience that there is no limit to how fast an object can move if you keep accelerating it is actually incorrect. Both position (in the direction of motion) and time measurements are slightly nonlinear (and given by the Lorentz "boost", see What is a Lorentz boost and how to calculate it? and https://en.wikipedia.org/wiki/Lorentz_transformation), getting increasingly inaccurate with increased speed. So the conventional fixed maximum speed of any object, which is the speed of light in a vacuum, is actually an infinite speed when properly measured. Obviously, it is impossible to reach an infinite speed, no matter how much we try to accelerate, because the energy required for acceleration to true infinite speed would be infinite. The conventional idea that the speed of light is fixed is only correct due to our mistaken way of measuring speed. If we did measurements according to how Nature actually works (see the Lorentz references above), the speed of light would be infinite. You can't go faster than infinite speed!
Second, one-way measurements of the speed of electromagnetic radiation from a near-point radioactive source to a spherical radius show no anisotropy.
Third, photons have been slowed down dramatically through laser refrigeration and other techniques, but measurements of the one-way (slower) speed of light show isotropy to within the fairly good accuracy possible in the experiments.
Fourth, the Global Positioning System relies on accurate timing to compute a location and elevation of any GPS receiver. The time values transmitted by the 24 current satellites are corrected by software separately for the special and general theories of relativity. If there were any error in either theory such as an anisotropic speed of light, the GPS would not work to the phenomenal and repeatable precision that it demonstrates every day, all over the world.
