soft question - What are examples of theorems get extensions based on simple observation?

Perhaps Euler's polyhedral formula:

V(vertices) + F(faces) - E(edges) = 2

provides an example of what you mean?

Euler did not give a proper proof but shortly thereafter this result inspired huge advances that had dramatic effects on the evolution of geometry, topology, convexity, and what today is called graph theory.

One measure of how rich this topic is can be seen from the many types and styles of proofs that can be found for this result collected below by David Eppstein:

http://www.ics.uci.edu/~eppstein/junkyard/euler/