Both Cavanagh and Pålsson stated that the claims were made by YouTube music network INDMUSIC, which Tune Core has partnered with to license music involved in its publishing deals. TuneCore's post explains that when artists sign a publishing deal with TuneCore, they are granted "the right to collect the royalties [their] compositions earn when they're downloaded, streamed, and used around the world." This includes usage on services like Spotify and YouTube.
The same post offers a step-by-step walkthrough for artists that wish to have their channel whitelisted so that it's not monetized by TuneCore and INDMUSIC. However, TuneCore notes that even if artists whitelist their personal accounts, "TuneCore and INDMUSIC will still monetize other people's videos on YouTube that use your music, and you'll collect that money.
Unfortunately, it sounds like Pålsson's desire to allow Let's Plays and reviewers to use his music without collecting revenue isn't compatible with TuneCore's publishing deal, even if he whitelists his own channel.