I was in the middle of typing out a few paragraphs about how it's ridiculous to think that (most) streamers are hurting music sales, taking advantage of artists by playing music, or that record labels need to do shit like this, but I think I arrived at a different, more practical idea, so I'll focus on that instead.
The reason we can mostly all agree that one shouldn't be able to use someone else's music, like in some blockbuster movie, for free is that they are making money from the artist's work, right? But how are they making money from that? It isn't because some people want to see Action Movie A and some people want to hear Pop Song B so they both pay for a ticket and sit down in the theater, the second group only there to wait patiently until their song comes on and then leave afterward. They make money because the use of the artist's music raises the value of the film, but good music isn't going to make a horrible movie good. Music raises value by a multiplicative, rather than additional, amount. If the use of a song raises the value of some blockbuster by 5%, as in earns them 5% more money eventually than they otherwise would have, then that's a huge load of money which would usually easily cover the expense of a licensing fee for the music. On the other hand, if the same song raises the value of someone's stream at the moment by 5%, that's like a dime or something (for the vast majority of even paid streamers), and yet to my understanding licensing fees usually have some huge upfront cost attached to them, they tend to take a while to go through, and there's basically no precedent for licensing music for game streaming so it may not even be available to streamers in many/most cases.
If streamers actually had a reasonable option to pay the copyright holders a sum based on the actual expected value worth for their stream earnings, I think most streamers would be fine with that. In a good system, a streamer would play whatever music they like and then when streaming payday rolled in there would be a certain small, reasonable portion of ad and subscription earnings for money earned during or shortly after the time when a given song was played and everyone could walk away paid and satisfied. Even this much I think would be very generous to the licensers because in almost any case if someone hears a song they like, usually amid gunfire and explosions and dick joke commentary, they'll want to hear it again, resulting in the free marketing that game publishers are coming to accept that game streams are, but even if we want to be as conservative as possible with copyrights, the record labels should be working on getting a system to streamers that actually allow them to pay for the music usage in a reasonable way instead of issuing takedowns that do nothing but hurt absolutely everyone involved, from the streamers to their viewers to the music artists to themselves.
Anything else is just the latest in a long line of tantrums of another old scared and confused media dinosaur as it's dragged kicking and screaming into what should have been reasonable policy a decade ago.