I don't know what alternative music is, I think it was first used with bands like Wham and Frankie Goes To Hollywood, bands that were not quite new-wave or pop before they broke. Over the last 30 years it's been applied to everything but mostly more basic music.
Locally here we have at least two stations flying the banner, the most famous is 89.3 The Current which is an NPR Station. They don't call it alternative anymore (sometimes the DJ's do), it's now known as independent music. 20 years ago they were breaking bands left and right, not so much now-a-days, some of it is pretty out there and hard to listen to for more than an hour.
As others have said, music as we know it died around the late 90's with radio deregulation, most mainstream alternative stations were bought up and merged with all the other local stations, competition was killed and non-competing formats were established. The largest buyer of stations, Clear Channel, also managed or owned most concert venues so guess what music they played the most? Metallica is coming to your town next year, is it a coincidence that they are played every 6th song?
The radio business model has changed, stations aren't valued or popular for breaking bands or news scoops anymore.