The "deeper theory" is that the universe is causal. In a causal universe there must be a maximum speed for communication; experiment teaches us that this is the speed of light.
It is possible to frame the axioms of Special Relativity in terms of causality; you will find papers and books on the various axiomatic approaches primarily in the philosophy of science/mathematical literature.
This has been discussed in Is causality a formalised concept in physics?Is causality a formalised concept in physics? and elsewhere.
