The trained AI model is a digital artifact that can be copied and distributed in a way that 'me listening to the music' cannot. The model contains detailed information about copyrighted content in a format capable of reconstituting infringing derivative works from a suitable prompt. It's 'really the same as' a huge content archive stored in a proprietary format with some lossy compression.
> The problem occurs if I copy the music.
Yes, I would argue that the 'problem' (copyright infringement) occurs when a trained AI model (essentially a big content archive) is copied and publicly distributed. For a hosted service, I would argue that the infringement occurs when the service copies and distributes an infringing derivative work in response to a prompt.