In these days of schools having websites, Facebook, Twitter and the ability to text parents directly, it does seem something of an anacronism to read out such a long list on air. That said, it does paint quite a nice picture of the situation across the station's patch, if all of the schools in one area are shut but they're open in another that tells you something.
On at least one occasion Radio Sheffield did an AM-only opt out for the closures list and carried on as normal on FM.
Do local radio stations have the resources to compile lists of schools these days? I believe at one time the arrangement was that the school phoned a specific number (not the usual studio phone in number) and quoted a password, as a safeguard against hoaxes. I guess just answering the phones could tie up a Broadcast Assistant/Assistant Producer for the whole show.
Write that down in your copybook now.