My ex made me watch Korean dramas with her all those years ago and now I have watched over 400 Asian dramas. As such, I have already watched most, if not all of the top rated and popular dramas. I need some new series to watch while I wait for this season's episodes to come out ever week. No real criteria but preferably Japanese or Korean drama. Historical dramas are ok but I binge watched them a while ago so I'm not that into the genre as much anymore. I especially like music centered dramas, so if there is a good one please recommend it to me. 

Just as a little piece of advertisement: for this season I'm watching Chief Kim, Defendant, Hwarang, Introverted Boss, Missing 9, Rebel: Thief Who Stole the People, and Voice. I like all of them, but Chief Kim is my favorite, so check it out if you haven't done so already.

you've seen way more than I have so there's a chance you've already tried these...

I definitely & highly recommend:
boku no yabai tsuma
juhan shuttai
white christmas (the 8-ep kdrama special. it def has major flaws but there are those of us who love it despite that.)

there's also... 
border
candy house
ishi no mayu
suisho no kodo (ishi no mayu sequel)
seirei no moribito
family's form
nigeru wa haji da ga yaku ni tatsu

some of these really aren't 'hidden gems'---I'm just going off of what I didn't see in your completed list, which I'm guessing doesn't list everything you've seen. I hope there's something here for you anyway.
Not music specific, but good underrated shows:  Evasive Inquiry Agency, Capital Scandal. Not really a hidden gem, but I didn't see it on your list and definitely about music - Nodame Cantibile.
I don't know if these are hidden gems or not, but these are some of the good ones that I really like, and would recommend:

- The End of the World (SK): psychological thriller x medical x zombie apocalypse
- Secret Investigation Record aka Joseon X-Files (SK): sci-fi, historical
- Ruler of Your Own World (SK): melodrama--the female lead is a struggling musician, if it's pertinent to your interest, it's not really about music though
- Merry Mary (SK): wacky comedy--since you like Chief Kim I reckon you might enjoy this
- Yume wo Ataeru (Japan): psychological drama
- Yami no Bansosha (Japan): crime drama featuring manga artists and editors
- Yakou Kanrasha (Japan): crime, family
- Namonaki Doku and its sequel, Petero no Souretsu (Japan): psychological drama, can be enjoyed as a stand-alone

Oh, and I know you're looking for dramas, but I can't help recommending these movie since you said you liked musical dramas (all of these movies are slice-of-life, by the way, so they're kind of slow):
- Linda Linda Linda (Japan)
- Misono Universe (Japan)
- Dangerously Excited (SK)
Thanks for all the recommendations so far, especially all the Japanese ones. I watch way more Korean dramas than I do Japanese ones, so knowing a few titles definitely helps. Also, I know I said dramas, but feel free to recommend some good movies as well!
I'm not really sure if these are well known or not... but some movies I could recommend trying (if you haven't seen them already) would be:
- Blue Spring
- Pluto
- Confessions
- Bleak Night
- Lychee Light Club
- Lesson of the Evil
- 20th Century Boys
- Okaasan, Ore wa Daijoubu

As for dramas maybe:
- White Christmas
- Jungle Fish 2
@Rawr123 I can't resist offering up some movie titles, so here I go

hero
fish story
a tale of two sisters
train to busan
bleak night
starry starry night
I'm a cyborg but that's ok
confessions (2010 jmovie)
the host
oldboy
sympathy for lady vengeance
sympathy for mr. vengeance
the good the bad the weird
nobody knows
memories of murder

again, not all of them are hidden gems but stuff I don't see on your list & are worth checking out IMO. 
Absolutely LOVED these recently:
Kazoku Game (Japanese) - psyche thriller.
When A Snail Falls In Love (Chinese) - cool police procedural, tiny bit of romance.
The Disguiser (Chinese) - wartime spy drama.
Love me If You Dare (Chinese) - serial killer.

I know you said Korean or Japanese but these Chinese ones are excellent in every way, beautifully filmed and acted. I've lost interest in Korean dramas lately. The only Korean one I'm watching is Innocent Defendant, and I'm loving it. I've just started a run of Japanese dramas, and am currently watching Siren, to be followed by Byakuyakou.