[Jaydelart]

I listen to Japanese music because I enjoy it, and it also, to some extent, helps me become more familiar with the language. I have learned several things from simply listening. That's several things more that I probably wouldn't have learned (as soon) otherwise.

I openly share that I listen to foreign music. If it's something I enjoy and it's not causing anyone any harm, I wouldn't let other people's opinions affect me too much.

If they ask me why I listen to foreign music, I simply tell them I'm learning the language. If they can't sense the correlation, there's no point discussing it any further with them; they're stupid.
... However, if you tell them you are learning the language, most of them will understand. At least, that's what I've experienced -- given they ask at all.
I'm not sure how it is for other people, but I believe immersion is just as important as structural learning, so it's perfectly logical to listen to music.


Also...
Generally, with music, I value the rhythm and sound more than the lyrics. Unless they are saying something significantly suggestive or offensive, I'm not too critical of what they're saying. Half the time, I can't even understand what they're saying in English songs -- much less in Japanese (French, Spanish, or Korean).