Two issues I've observed.
The most recent California earthquakes captured the news a bit nationally — over the holiday weekend. Unfortunately, I think three "inappropriate" lessons were being taught by these twin events.
1. If you ask people where in the United States there is a significant earthquake risk, the immediate answer will be California. Thus, having this event "in California" reinforces that line of thinking. I found a bit of satisfaction in that the governor of California Gavin Newsom did talk about "a wake up call" for Californians and "others" with an earthquake risk. Still, most people feel secure, in that earthquakes — while possible and potentially significant — are rarer in other areas of the nation.
2. People, in speaking with the media, spoke about "running out of the house" or building — the absolute wrong thing to do in an earthquake to avoid being injured. People hearing these reports from "survivors" will have their "fear and flight" mechanisms reinforced. Drop, Cover and Hold!
3. The second negative lesson was that the amount of damage inflicted by the earthquake was pretty minimal given the magnitude of the event. Again, if this event had been in an urban area and not one that is basically rural, the damages would have been much greater. People likely are thinking, well it was a "BIG" earthquake, and the damages are not that bad.
Which means there will be more work to be done to capture people's attention to the earthquake risks when you try to promote earthquake readiness.