Answer:
First, chlorhexidine (CHX) is available by prescription only (in the U.S.).
Also, CHX has a tendency to severely stain teeth when used daily. The stain can be removed during a professional cleaning, but it can be quite distracting in the meantime.
Finally, CHX tastes absolutely terrible. Most formulations have taste masking agents, but they are not all that effective. If water or other liquid is swallowed following use of CHX, the masking agent is washed away, and the taste of CHX becomes even more pronounced.