New equipment

It may be interesting to note that all of the old, established, time-tested components in our systems were "new" once.

Unfortunately, it isn’t so clear cut that you listen to your system OR your music.
As pointed out, the most desirable is to listen to the music first and then everything will fall into place. However, once you’ve been bitten by the audiophile bug, your enjoyment becomes wrapped up as part of a sonic/musical experience. You listen for BOTH music AND sound. That is what makes this hobby so  precarious. The trick is to not let the sound be the prime mover.

I can and enjoy doing both.

I'd have to agree with slaw in that I can enjoy both as well. Likely every listening session of mine has a bit of the analytical as I'm still experimenting with entire streaming setup, changes constantly ongoing so my attention is purposely directed there. I think the problem with analytical listening is when one doesn't want to listen in that mode. IME, that indicates issue with system, or one may have severe case of audiphilia nervosa.

Quite often audiophiles will buy a component because it enables them to hear “things in the music they have never heard before”. This can sometimes be a real trap if you aren’t an experienced listener.

Quite often it is the result of an unnatural presentation of the music which skews frequency response, leading edge presentation, imaging, ambience, decay or some other parameter which exaggerates certain parts of the musical experience. The listener then becomes focused on listening to those new exaggerations or “highlights” in the system and the way they impose their signature on the signal, rather than the music as a whole. Some amps ams and speakers are designed to be really good at spotlighting certain aspects of a musical performance, which may be initially appealing but fall short in being able to offer a balanced presentation which satisfies long term.

Any new information presented has to be taken in context: does it genuinely add to the musical experience? Does it distract from the flow of the music or does it integrate? Does it sound natural or artificial? Is it merely the result of existing information being highlighted? Is other information being obscured as a result ? Does it make you want to play more recordings? Does it help make you forget you are listening to recorded music?