That's one definition of a CTO. Another CTO type is the opposite: "the thing you call an engineering founder when they've done so much customer-facing work that you have to take their commit bit away from them". This is, I think, an even more common archetype than the other one.
Then you have the toxic CTO definitions --- CTO as "ultimate decision maker for engineering", or, God help you, CTO as "executive manager of all of engineering".
You'd have to be specific about what kind of CTO you are to really make the question of why you code interesting.