Taken from a previous answer:

In the recording industry in the 1950s (or 1920s, for that matter), pretty much every popular musician covered every vaguely appropriate popular song. The Rodgers and Hart tune 'My Funny Valentine', for example, was covered separately by both Chet Baker and Frank Sinatra in 1954, and separately by both Ella Fitzgerald and Miles Davis in 1956 (and those artists are not out of the norm). And 'My Funny Valentine' is probably one amongst a dozen songs that each of these four artists each covered in this time period.

In fact, I have a book of Australian charts from the 1940s to the 2000s, and before the late 1950s, the charts in that book were focused on the song, rather than the performance - all of the different acts that were getting airplay/sales were listed after the name of the song, rather than each performance of the song being listed separately. And so one thing to remember is that the music industry existed before the recording industry; the primary way in which Tin Pan Alley songwriters made money is by selling sheet music, by getting people to play the songs in their own homes. In this time, there was a job called 'song plugger' which was basically the job of convinced live performers to introduce new songs into their repertoires so the song would become better known and thus sell more sheet music. So what changed between the old era of song pluggers and every artist covering every song and the current era of Katy Perry's singles all being new songs?

One obvious change is that, by the mid-sixties, the way that people thought about records had fundamentally altered. In 1956, the record was, literally, a record of a performance - Frank Sinatra's version of 'My Funny Valentine' was likely recorded in one take on a stereo microphone setup, with Sinatra, in order to be louder, simply standing closer to the microphones than the orchestra.

But by 1967, you have something like the Beatles album Sergeant Peppers Lonely Hearts Club Band, assembled by overdubbing new performances on top of old, full of electronic effects and manipulation of tape recordings; it's a record that could not have been performed live at that time. By Sgt Peppers, it was clear that the record was no longer a record but an object of art in of its self. And if the record itself, rather than the song, is the object of desire, then a cover of a familiar song is no longer quite as much of a crowd-puller as it had been.

Going along with this change is that the ethos of jazz and Popular Music (in capitals) was replaced in the 1960s with the ethos of rock music and pop. In jazz (and/or the adult pre-rock Popular Music epitomised by Sinatra), individuality was demonstrated not in the choice of song, per se, but how that song was used as a canvas for virtuoso self-expression - in how Miles Davis soloed over the chords of the song, or how Sinatra chose to phrase the melody. In contrast, in rock, individuality is most strongly demonstrated through having a new song and sound of your own, preferably that you wrote yourself.

So these days, when Ryan Adams covers an entire Taylor Swift album, he is specifically covering that album - he is recording the covers in the knowledge that his is a secondary work, that people will be comparing his work to a more definitive work. In contrast, all of those versions of 'My Funny Valentine' in the 1950s were attempts at reaching the unreachable Platonic form of 'My Funny Valentine' epitomised by the sheet music, with none of the performances truly being definitive. 'You Belong To Me' belongs to Taylor Swift in a way that 'My Funny Valentine' can never belong to Chet Baker.

And many of the artists you specifically mention from the late 1960s as doing covers make styles of music which were in some ways more influenced by the ethos of jazz than the ethos of rock. Burt Bacharach and Dusty Springfield both come from a sort of easy-listening, light-entertainment background which had the values of Popular Music rather than pop, because their music was often aimed at older audiences who had grown up with the older values of Popular Music.

Similarly, there was a multitude of covers by soul artists in the 1960s - for example, Sam Cooke's 'A Change Is Gonna Come' was covered by Solomon Burke, Otis Redding and Aretha Franklin, amongst others. This comes from a style of music with big influences from jazz; most of the Funk Brothers who were the band on Motown records, for example, were jazz musicians 'slumming it'. It wasn't until the turn of the seventies that you get soul/funk records like Marvin Gaye's What's Going On or Curtis Mayfield's soundtrack to Superfly where the ethos of rock, with its focus on the sound of records and on the statements being made in the music, becomes a more dominant flavour of black American music. It's not coincidentally around this time that the great tradition of the soul cover of the recent song begins to dry up.

I do feel covers of recent songs aren't taboo these days - just look on YouTube and you'll see a million covers of any recent pop hit, which suggests that there's a market for them - but the cover definitely doesn't get a talented pop artist as far as it did in the 1950s, because of the change in ethos and the change in the way that we listen to recordings now.