Conservation of angular momentum really is a new phenomenon, one that does not follow from the Newtonian mechanics you already know; therefore it deserves its own place as a law. Specifically, you have proven that
If a system experiences no torque, then its angular momentum is conserved.
However, this statement by itself is useless. Maybe all systems always experience torque; maybe a system can exert a torque on itself. What we really want to say, i.e. the actual law of conservation of angular momentum, is more like
An isolated system's angular momentum is conserved.
To see how these are not equivalent, suppose we have a system of two isolated particles, one above the other. Newton's third law does not forbid the particles from pushing left and right on each other. But then the system will begin spontaneously rotating! It changes its own angular momentum by exerting a torque on itself.
To force conservation of angular momentum, we need to use the strong form of Newton's third law,
Forces between particles come in action/reaction pairs, and these forces are directed along the line of separation between the two particles.
This is a fundamentally new assumption, so angular momentum really is its own thing. On a deeper level, conservation of linear and angular momentum follow from translation and rotational symmetry of space, and it's possible to have spaces which only are translationally symmetric, or only are rotationally symmetric. The two are independent.