The thing is, if you're planning a coup with military backing, you don't do it when your most rule-abiding, law-respecting general is in town. You send that guy to guard the Alaskan border or whatever, and instead recruit a general who's a maverick rules-breaker and who gets on well with you personally.
So the military should respect the constitution, but when it comes to a coup you'll get whichever general respects the constitution the least.
I don't know much about coups actually work, but a general does not make an army.
A general that wants to stage a coup seems like they must still require the support of the troops.
Speaking anecdotally, every unit I've been in not a single man would follow the questionably illegal orders of any general unless they had full respect and confidence in that general, and typically the troops only have full respect and confidence in a subset of their immediate leaders (which are not typically generals). I guarantee a LARGE percentage of troops would treat the highest ranking general as an enemy combatant if their direct (low ranking) leaders who they respected convinced them that the general's orders were illegal or against their oaths. Soldiers don't die for generals, they die for each other, and "each other" is usually enlisted or low-ish ranking officers (maybe captain and below in the US). A professional and disciplined soldier will charge a hill risking certain death on the orders of a general, but a professional and disciplined soldier will not stage a coup on the orders of a general alone.
I agree that soldiers don't stage a coup on the orders of a general alone.
You need a general with likeminded officers, and a convincing excuse for the rank and file to go along with their officer's orders.
Something like "the election was stolen, the winners weren't legitimately elected, we've got to defend our country". It doesn't need to survive detailed scrutiny, a few hours is long enough for the major scrutineers to accidentally fall out of windows.
Said military was also filmed live with empty pistols and training magazines on chest rigs, so it was clear that they were never on the President's side. SK has mandatory military conscription and so implication of waving around scary but empty guns was immediately understood and shared by local citizens.
I'm not an expert on South Korean constitutional law. But from the parts that others have quoted here, if the legislature declares an end to the martial law, that's the end of the martial law. The military should not then be obeying the president's orders to impose martial law, because the martial law is over.
In the US, military officers take their oaths to obey the constitution, not the president. I don't know if that's true in South Korea.
The Korean Constitution says that once the legislature declares an end to martial law, the President "shall comply". The military has to obey the President's orders until that time though, and 'shall comply' has two flaws: (a) it doesn't contemplate what happens if he doesn't and (b) it contemplates time passing, but doesn't specify the duration. These flaws are moot, because the President has complied.