> TV channels and radio stations will just broadcast a breaking news thing.

Many young people don't even own a radio or TV any more, and even more own one but don't use it regularly.

> Websites will have notices

That assumes that a) the Internet is still up everywhere (by the way, same problem with the apps crap) and b) that people are 24/7 surfing on news websites.

> I'm entirely unclear why you'd need an emergency broadcast system? Or place so much emphasis on its importance?

To announce there is something going on and people to have a look at the TV/radio for more information. That's all.

> And why would having one that worked over AM or even FM be worth anything? People don't buy those radios any more. What difference does it make?

AM radios are incredibly cheap to build - all you need is a loop, a tunable capacitor, a diode, a fixed capacitor and an earphone. When I was a kid I had one as a "build it yourself" kit for children. FM radios are more complex but are literally everywhere - many cellphones have them, virtually every car built after the early 50s, kitchen appliances, bathroom radios, classic radios... DAB/DAB+ radios are relatively new, in fact they are only required to be present in cars after end of this year. And in contrast to AM/FM it is almost impossible to build one out of basic components yourself.

> What you're describing sounds like a massive white elephant that was probably worthless even in the cold war. A proverbial safety blanket.

There's more need than just a war for an EBS. Stuff like "the power plant nearby is leaking radioactive gas", "the industrial plants are on fire, close your windows", "there's an earthquake predicted, cover yourself", "flooding immanent" or "active terrorist case with a shooter on the run, stay indoors" is some things that I can easily imagine.

In case of, say, fire at a chemical plant, it may be important that people downwind get notified right away, there might not be very much time to get inside, close windows etc. Same for collapsed dams, radiological accidents, that sort of thing. Sirens can do that very well. Other channels are much less reliable (people might not have a news website open on their computer, might not be looking at their phone to catch that notification, etc.)

Radio is important because it's so widespread and low-tech. Pretty much every car has a receiver that can run for ages, there are inexpensive emergency ones that can be powered by crank and solar, so even in prolonged power outages, as long as the stations remain powered, and someone in your neighbourhood has a working receiver, it's still useful and pretty low-cost, since the infrastructure is already there.

FM is still widely used, in most European countries still way more than DAB.