They tried. They failed so horribly that the RIAA basically granted Apple a monopoly on digital music for the next half a decade. Steve Jobs had to basically slap the labels around a lot just so that they wouldn't limit how many iPods you could sync your music to or make you buy a subscription. Music streaming was tried by basically every other player in the business and failed until Spotify pushed for ad-supported free streams.

Interestingly enough, the record labels - or at least Sony and Universal - actually do have minority ownership in Spotify. I would not be surprised if that was specifically pushed to get those labels on-board with Spotify's really low royalty rates.

As for why they don't do it today... my guess is that the exclusive content model just isn't as much of a draw for music as it is for movies and TV shows. Or, more specifically, we don't listen to music like we watch TV shows. Imagine if someone just made a list of episodes of Netflix shows they liked and watched them in a random order - that'd be insane. But that's how we handle music all the time. Pulling your music off Spotify means you lose out on radio and playlist royalties, because nobody can make playlists across streaming apps. You'd have to switch into the one streaming service that has the latest Taylor Swift album, then switch back to the one that has the Sony BMG catalog, and so on. Nobody would do this.