It actually depends, but most of the time, yes, you would be violating PLA/TOS and YouTube Terms of Service if you played a song unless that song was explicitly listed/licensed under a license that is “royalty free” for public/broadcasting, Creative Commons, or any other permissive license.
Will YouTube/the creator pursue legal action? Probably not since it’s such a small/hard to identify violation.
Could they? Yeah they could.
Considering the server provides the function to play the music, the server owner would need a third-party authorization/license, like a business would buy to play music in its business.
That violates TOS/PLA, since third-party licenses like that are not allowed. I asked a similar question back after the acquisition.
The software itself (playing YouTube videos/music services) does not violate TOS/PLA by existing, but it is more than likely being used to do so. That’s (probably) why you don’t see music players on Releases anymore.
For a music app to be 100% in the clear, you’d probably have to make it so that it ONLY plays music from verified, CC/unrestricted licenses but even then you might run into issues with YouTube and their ToS. For example, maybe an app that only plays music from a verified royalty-free source or music in the public domain could work.
Most music people want to listen to aren’t from either of those.
Edit: From the PLA:
5.6. Limitation on Musical Works and Modification of Voiceover Performances. You agree that you
will not use the Creator Services to create, upload, or distribute any musical work or sample thereof that
is subject to license by a third party including, without limitation, any performing rights organization
such as the American Society of Composers, Authors and Publishers (ASCAP), Broadcast Music, Inc.
(BMI), SESAC or any similar international organization responsible for the licensing of public
performance of musical sound recordings and/or compositions. For clarity, from time to time the
Company may incorporate third-party musical works into a Covered Game, but nothing in this Creator
PLA gives you the right to publicly perform, copy, distribute, sample, remix, or otherwise modify or
exploit such musical works as part of your use of the Creator Services. You agree that you will not use
the Creator Services to create, upload, or distribute any Server Content or Player Content that includes
modifications or alterations to any voice recording, voiceover performance, or element thereof
contained in a Covered Game.