Why is the speed of light constant for all observers?

[Ambitwistor]

From what I understand of SR and GR, when we dig deep enough we find that the constancy of C for all observers lies at the base of both theories. Can we say why C is constant for all observers? Is there a more fundamental statement in this regard?

Not really. There are other formulations of relativity in which the "speed of light is constant" axiom is derived, not postulated ... but you can always play that sort of game: pick a few theorems of a theory, take them as axioms, and derive the old "axioms" as theorems from the new axioms (previously theorems). It's not really anything deep that goes beyond what we already know. But it depends on your sense of aesthetics. For instance, if you really don't like instantaneous action at a distance, then you might take causality axioms as fundamental, and then that would "explain" the constancy of the speed of light. (In non-relativistic theories, effects can propagate at arbitrarily high speeds; if you set a speed limit, then you can assume some additional symmetry principles that force the speed limit to be universal for all observers.)