Some countries already do this! For example, the German LF time signal, DCF77, dedicates some of its "spare bits" to broadcast civil protection and weather for

Some countries already do this! For example, the German LF time signal, DCF77, dedicates some of its "spare bits" to broadcast civil protection and weather forecast information, the latter of which many simple LCD "home weather station" thermometers can receive and decode. GNSS (navigation satellite systems like GPS or Galileo) are also an obvious candidate for this, given that essentially all phones already can receive their signals.