Relative speed

People say light always travels at speed c.
Light always travels at speed c relative to an inertially moving observer. (Or to an inertial frame that is momentarily moving with your observer.)
People say that nothing can go faster than c
Nothing can go faster than speed c relative to an inertially moving observer. (Or to an inertial frame that is momentarily moving with your observer.)
The distance between light moving in two directions grows at speed c
Each beam of light is moving at speed c relative to the person on the ground. So the first rule is fine. And if someone could go at speed c then we might have a problem, but no one ever claimed observers can move at speed c relative to another observer.
So we can actually conclude that inertial frames can't move at speed c relative to an inertial frame and so therefore neither can observers.
Your result demonstrates that not only is light speed a maximum speed for anything, it is a speed that observers can not achieve, they must always go less than c.