The details of the case, however, may be less important than the lawsuit’s real purpose: To repeat the music industry’s familiar refrain that digital music companies like Spotify and Pandora don’t value songwriters, and demand courts and lawmakers step in to give them more money.

Think this is an exaggeration? Then, check out the opening of the New York Times’ account of the lawsuit, which reads like a propaganda piece for the music industry:

When Michelle Lewis, a Los Angeles songwriter, gets her quarterly royalty statements from Ascap, she receives a stark reminder of how songs are valued in the digital age. The tunes she writes for TV shows like Disney’s “Doc McStuffins” bring in thousands of dollars, but streaming outlets like Pandora and Spotify yield less than $100 combined.

“The honest truth is that if it weren’t for the TV stuff, I’d be working at Starbucks,” said Ms. Lewis, who has writing credits on pop hits by Cher, Little Mix and Katharine McPhee. “There is no way I could afford to be a songwriter just on streaming and digital radio.”

The anecdote sounds devastating. After all, something must be wrong if tight-fisted Pandora could force poor Ms. Lewis to sling lattes rather than write songs. And indeed, something is wrong—but not in the way the story suggests.

Yes, the rise of the streaming music economy has hurt musicians, but Spotify and Pandora are scapegoats more than they are robber barons. Both companies are losing money and have long been beset by nasty and expensive lawsuits driven by the music industry.

Instead, the problem lies in how copyright law allocates royalties between songwriters (the people who write the songs) and performers (the people who play them). As it currently stands, performers can get paid as much as 14 times more than the songwriter when Pandora streams a song. And don’t forget that when a traditional AM/FM radio station plays a song, they pay about the same amount as the digital stations to songwriters—and nothing at all to performers.

If you want to make an analogy, you could compare it to a restaurant. Imagine, for instance, there are laws that say chefs and waiters get a cut of whatever a customer pays for a meal. Would it make sense to pay the waiters 14 times more than the cooks? Probably not, but you wouldn’t blame the owner of the restaurant for that—and that’s the predicament in which Spotify and Pandora find themselves. They pay the majority of their revenues towards music royalties, but then take flak because the law distributes it in an odd fashion.

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As for the Times’ comparing the payments made by Disney (DIS) to those by Pandora and Spotify, it’s an apples-and-oranges situation involving two different types of copyright. (The rules for buying rights to a song for a TV show or for a CD are very different than they are for radio stations and always have been).

But all this is complicated. If you want to get deeper into the weeds it’s extremely complicated. That’s why the music industry prefers to tell simple stories about tech companies robbing musicians. And why not? The tactic has proved effective since the press regularly falls for it, and so do many politicians who are star-struck by the one-sided accounts of celebrity musicians.

To be clear, this doesn’t mean that songwriters don’t deserve more. They probably do, and it would be comforting for fans to know the artists they love are making enough money to keep writing songs.

Fixing the system, however, will take more than lawsuits like the one filed this week. (The case is about how the Justice Department oversees so-called consent decrees, which ensure that royalty collection giants ASCAP and BMI can’t wield monopoly-style powers.

Instead, sorting out songwriter payments should involve changing how payments flow to performers versus songwriters, and perhaps also requiring traditional radio stations to start paying performers too.

Finally, if the music industry really cared about young songwriters, it would offer to roll back the greedy copyright changes it has obtained from Congress in recent years. Those changes involved extending copyright further and further back in time, so that a large share of songwriting royalties are now paid for tunes written 50 or more years ago. Just imagine if copyright terms were shorter, and more of the money collected went to new musicians instead of aged or dead ones.

Alas, the chances of any sensible copyright reform appear as slim as ever given the bad blood that exists between the music and tech industry. But if there is going to be a debate, let’s at least start with the right facts.