Why did he declare martial law?
Because he lost the election.
To hear him say it, SK has a deep state undermining him. To be honest, I didn't take time to study how true it is.
But given what happened to Trump, and given, as the original poster asked about, that the media has conveniently forgotten this very important topic, it may be that the U.S. media doesn't want us to realize how common it is for nations to have a deep state undermining their elected officials.
Because he is an idiot. He will either resign or be impeached in a few weeks. This is a nonideological problem. If martial law persisted, there would have been a military coup. Yoon did not do his 2 years of military service. At some point, the generals would have wearied of him.