One of the ways some record labels got around that in the US was to have their artists record songs published by a company owned by the label.
That's part of it. In addition, when they sign artists, if that artist writes, his songs are published by the labels publishing company. Plus, there are 360 deals, where labels get a portion of their artists merch or touring revenue. That is never mentioned in any of the discussions about the radio royalty. This issue of labels wanting money from radio goes back 60 years to Sinatra. But in that time, the labels have found ways to get a portion of that money without having the legislation. In other words, as you said, radio is already paying them the royalty through the publishing. If they get their label royalty, they won't give back the portion they already take from the artists. The new money will just be added to the publishing money they already make.
This has become a bigger issue because of streaming. The labels want to equate radio with streaming. That's what they said in the congressional hearing last year. What they ignore is that streaming is digital, and that's why they get the royalty. Broadcasting is analog and not covered by DMCA. They want broadcast radio included under digital copyright without allowing it to become digital.