I think doing it the way they do it now is the most effective way to get any widespread emergency or otherwise urgent message out to the public.
Where I live, I don't have to worry about it, but there are still places in the country with slow Internet connections who might not be able to easily access a news website, especially to watch a live streaming video. More people are more likely to at least have a television and an antenna.
Additionally at least for me, I am not always where I can get Internet. Sometimes I'm at the dentist office, for example, where there is not Wifi available for waiting patients to use. So if I see that message on the television in the waiting room telling me to visit their website, I'm out of luck as I don't have a data plan for my phone or other devices.
And then there's the fact that their website, CBSNews.com in this case, might not be able to handle so much traffic all at once, forcing the website down and forcing their viewers to choose a competitor's website. With television, there's no danger of the television signal going out if too many people trying to watch at the same time.
Those are my thoughts on the subject. I know everyone has their own opinions and that's healthy to get some debate going!