Over here in the Netherlands, we have had this system since 2012. I’ve never received a real alert, only test alerts. Tests are twice a year, and I think that’s a good balance between testing often enough to have awareness and irregular enough that it isn’t a real nuisance.
The system is mainly used for big fires with dangerous smoke. The alert then advises to stay inside and switch off ventilation. Besides that, I can see they also sent an alert a few months ago for a major power cut which led to people trying to call the emergency services.
So I would say the system as used here works really well. It all depends on the criteria for sending alerts, but I would wait and see before saying it’s a bad idea.