The Theory of Special Relativity is based on the postulate that the speed of light in a vacuum is always $c$ in any inertial frame of reference; this is known as the second postulate of Special Relativity. This postulate is taken for granted and not proved, this is what postulate means. From the two postulates then we are able to prove a lot of amazing statements, including the beloved equation:
$$E=m \gamma c^2 \ \ \ \ \ \ (1)$$
Note that the presence of $c$ in this equation is not a coincidence, it derives from the proof of it, proof that can be found in any book about the subject.
But what if the postulate is not correct? What if the speed of light is dependent on other factors such as the one you mentioned? Then Relativity as we know it breaks down and we have to replace it with some other theory. A crucial point though is that the new theory has to be in agreement with the experimental results, so the new theory must imply pretty much all the phenomenon that Special Relativity predicts up to the current level of precision of the experimental data.
But in particular the hypothesis of the video you linked, abut the mean velocity being $c$, is special; at a first glance it doesn't seem to break much; this is because in pretty much all the proofs of Special Relativity we work with the mean velocity (in the sense of propagation forward and propagation back) of light, and so all the proofs remain valid even if the second postulate is modified in such a way. So even in this case you would be able to prove $(1)$, and just as before the presence of $c$ would not be a coincidence.
But there are two big problems with the hypothesis of your video:
The first one is that pretty much all modern physics is build upon the assumption that the universe is isotropic: there is no preferential direction in space. The cited hypothesis would break this fundamental assumtion and probably would cause a lot of troubles in many areas of physics.
The second problem is that the premise of the video you linked seems suspicious to me: the main statement is that we can't measure the one way speed of light because we can't be sure of the synchronization of two clocks far apart, why? Because of the time dilation effects of Special Relativity! Seems circular reasoning to me. You want to use Special Relativity to disprove Special Relativity. The arguments present in the video should be refined to avoid this problem.
But on top of this, leaving aside the problem of circular reasoning, in principle we can be sure of the synchronization of two clocks! We can synchronize them while they are together and then move them apart really slowly. The video you linked mentioned this method but stated that having a different value of $c$ in one direction complicates the matter. But in all cases we can be sure that the time dilation effect would be proportional to the relative velocity, so we can be sure that if the relative velocity is infinitesimally small then the time dilation must be infinitesimally small as well! So in principle we can synchronize two clock far apart, and the one way speed of light can be measured.1
: To be honest I am not completely sure of this last bit of reasoning: maybe there is a hole somewhere that breaks my statement that time dilation must be proportional to relative velocity. In any case still really suspicious video.