sorry when you said playing in your first paragraph that also extends to performance of said song? since you use the words playing and then later using and performing
and the only entity that has the right to allow or deny said playing/performing rights is KOMCA not ADOR (in relationship to the copyright issue only)?
There are 2 types of copyright owners when we are talking about a song: there's the author of the song (songwriters, composer, lyricsist, etc) and then there's the owner of the recording of this song (or as people call it, the masters). Authors usually work with publishing companies, while record labels are usually the owners of these masters.
When you are doing a live performance, in theory, you don't engage with the recording of that song, so you would only need to pay the authors. However, if you are at an event where the recording is being played (like a club, a bar, or I'm assuming a political event), then you will have to pay the owner of the recording too (I think the US is different in this aspect from europe, or maybe just for radio, I'm not sure). This is what its called a public performance right.
Now if these negotiations where to be done individually, can you image the amount of the licenses a single event will have to negotiate to be able to play/perform a single song? let alone a bunch of songs with different copyright owners? This is why performing rights organizations (PRO) exist, they are the ones in charge of giving these licencenses, collecting the money and then paying the copyright owners. Each country can have one or more PRO, this is what KOMCA is. Usually there's one association for authors and one for record owners (I think this is KMCA in korea). These associations are not just for licenses though, they can also advocate for the industry in general, as collectively they have more power.
This event is not in korea, so KOMCA is not the one giving the license. According to wikipedia HK's PROs is called CASH. In theory, CASH will collect the money and then give it to KOMCA, who would then pay the copyright owners. This should not be confused with the artists, they will get their own money for the performance.
Now imagine you want to make an ad or a film using a song from NJs. Again, you will need a license for both the authors AND the record owners. In this case the PROs won't get involved. You will have to contact the publishers and the record label individually and BOTH need to giver permision for you to use that song. This is where ADOR can say no and maybe HYBE publishing, since I believe they hold the rights for some NJs songs.
Now you are asking what if trump did an event, got their license and hire a cover artist to perform your songs or played the recording. In theory, there is not copyright infrigment there I believe. However, you can come for them for other things, like using your likeness for the campaign or stuff like that I suppose. However, if trump use the melody of your song and put his own MAGA lyrics to it, THEN its copyright infrigment, because to modify a song you absolutely need permision from the author. If Trump sampled your song for his own MAGA rap song, then they would need both the author and the record owner permission.
This is why I'm saying, ADOR probably will try and stop NJZ perfomance one way or another, but it won't be through copyright.
Omg this came out so much longer than I intended to, sorry >.<