> 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.