[quote=Dervish]
There's still a lot of music that does that, but you're definitely right; a lot of music now adays (and probably in the past) treats each song as a single instead of a cohesive story. I'm not one to subscribe to the whole "old music is so much better than new music" argument because the stuff you hear from previous generations is the stuff that , the vast majority of it forgotten. Every generation has good artists, every generation has bad ones. Every one has innovative trend setters, every one has mass produced sell-outs. It's actually interesting listening to the radio and seeing what songs from my youth are now the common go-too songs that are like the big ones on say a classic rock station. After a decade, it's still getting regular air time verses the sea of shit that wouldn't be around if it weren't for the Internet.
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I think it is more prevalent now, because the change was brought on by modern changes in technology. When people used to buy vinyl albums, it was expensive, and you'd really have to pick your favourite bands and stuff to buy. When you listened to it, you'd put on a whole album. But the way we listen to music has changed, as you noted - and that's what's brought on the change. Its not a matter of old being better versus new, just that the music has adapted alongside the manner in which we listen to it. 
I agree with the point about music surviving - it's called survivorship bias, I think, Vsauce (really interesting videos on YouTube, check them out) mentioned it in his last video. There was another interesting example he gave - when planes came back from battle in WWII, extra armour was added to the places with the most bullet holes, the logic being that they were the areas most targeted and most at risk, but they completely missed the fact that it was the survivors who suffered most damage there, and the armour needed to go everywhere else - i.e. the ones who didn't make it back were the ones who were hit where the survivors weren't.
I'm starting to experience that now - songs from my childhood that are now my generation's classics. Things like Busted and McFly coming back for a big reunion tour... shit, I was listening to them when they were originally together, and now they're having the famed big reunion tour every major broken up band has. Wut. Realising Doctor Who restarted in 2005 and that my favourite over childhood game, Pokemon Ruby, came out in 2004 or some such made keep feel ancient.