There is one HUGE difference between the two systems.
In your system where the speed of light is 1, can you measure changes in the speed of light?
What if the speed of light told you something extremely important, such as when the speed of light goes up, the stock market will go up, and when the speed of light goes down, war breaks out? (The speed of light is actually more important, but i don't have a better analogy)
So, measuring the speed of light is very important.
Assuming that the speed of light is constant, and so making a rubber ruler to measure it keeps you from ever knowing this.