Governments are recognizing that there are times when incidents stretch their resources to the point that having civilian volunteers involved is a positive, and that it is a good thing to train civilian volunteers in advance.
One of the key things CERT training includes is the incident command system, a widely adopted set of organization principles that allows not just first responders but also civil officers and volunteers to work together across agency lines efficiently. When incidents happen, fire and law enforcement agencies already know how to work together, and CERT volunteers are also trained how to fit in. This will expose CERT volunteers to principles helpful in other facets of their life (e.g. "scope of command," meaning don't be managing too many people directly, etc.)
CERT training is great prepping. There is no obligation to leave your family or neighbors just because you're a CERT member--in fact, the understanding is your primary responsibility is to your family and neighbors (this relieves pressure from first responders). Plus, you get plugged in to communications channels that can alert you at a very early point to possible threats. Finally, your CERT ID and perhaps even personal recognition means officials are more likely (not required, but more likely) to respond helpfully to your questions in some scenarios.
For example, some months ago the police and fire departments had barricaded a block near my home. Several neighbors said the police would only tell them to stand back. I held my CERT ID up and approached a squad car and asked what was up. The officer took several minutes to brief me on a potential chemical spill. (CERT volunteers get training on different types of chemical threats and we were issued a reference book that let us know such things as how far away is safe, where to establish perimeters, treatments for exposure, etc.) I was then able to explain what was happening to the neighbors standing around. I think this was CERT working as intended.
CERT volunteers are not meant to be enforcing laws or fighting fires, but during a "situation" CERT members can free up law enforcement and fire fighters to enforce laws and fight fires who might otherwise be directing logistics, disseminating information to neighborhoods, entering data in computers, etc. My observation is that the typical CERT volunteer is middle-aged, not real physically fit, and has a lot of spare or flexible time.
Sometimes CERT members are official liaisons with other groups; in my CERT academy was a representative of one homeowners association and a large church that was preparing to serve as a shelter for evacuees from wildfires or nuclear accident. These people planned to serve their primary organization by interfacing with city, county, and state officials in an appropriate and helpful way.