Are there any dramas where the main character is the villain? Or perhaps a drama where the main character (protagonist) loses and the antagonist wins?

i think you would like  Lucifer

Mmmmhh not sure if this is what you are looking for but, try with Vincenzo (2021) it's available in Netflix.
The MC Vincenzo  Cassano (Song Joong Ki ) a korean gangster from the Italian Mafia.
I loved how he played the MC in Reborn Rich it was the perfect adaptation for that Manwha.

Possibly Money Heist: Korea - Joint Economic Area? They're more anti-heroes/antagonists than villains, though.

You can watch Mouse, starring Lee seung gi. 

There's both a movie and drama but i suggest you watch the drama. It's much better than the movie and also nice details to it. 

When the Devil Calls Your Name. The drama is honestly pretty amazing. The MC is not a villain per se,  but is a terrible character who will leave you with a sleazy feeling. It was the first time I hated a MC and yet was undeniably charmed by them.

Also as this poster said, Mouse is a great drama for an all out villain among the MC casts.

 Mi_so:

You can watch Mouse, starring Lee seung gi. 

There's both a movie and drama but i suggest you watch the drama. It's much better than the movie and also nice details to it. 

Hard to say, since you didn't specify if you want romance in it, or not and how big of a villian you want the MC to be. Since I don't know, I will just state my personal favorites :)

  • KinnPorsche (mafia romance)
  • Duelist (assassinXpolice officer romance movie)
  • Flower of Evil
  • Kill It
  • Long time no see
  • Love between fairy and devil
  • My Journey to You
  • Recalled (great mystery movie)
  • Vigilante
  • A Killer Paradox
  • Till the end of the moon
  • V.I.P (killer movie)