I’m want a show where the male lead is cold with background trauma that affects the plot. I bonus points if they were some kind of soldier/ agent.

examples include 


Fiery priest

Moon lovers


The wolf

Love me if you dare


Any country is fine.

Money Flower - this is definitely the best korean revenge drama. Amazing storyline, amazing unique characters. Hits all your marks.
Burning ice - now here you have 3 males that would fit the bill. Absolutely amazing drama, highly recommend it.
Watcher - So here you have 2 male leads where one is questioning the others involvement in his mothers death. Neither is very warm in this drama, but one especially is very twisted and grey. Great drama.
Trinity of Shadows - this one is sooooo darn good. There is a male lead that seems more coldish, but the colder one of the group is probably the female lead. Then there is another male lead who is more open. All three of them have traumatic things happen to them. You named a lot of examples with romance stories, so this one would also fulfil the romance aspect. It really is a golden golden golden drama that I can never recommend enough.

Also a drama that you might really like based on the examples you named:
Pinocchio - I wouldn't per say call him cold, but I also do not call the K2 male lead cold. So he is more similarish to him there. Plus trauma story, forbidden romance, hidden past, etc. You will probably like it a lot.

2 dramas that are fitting cold leads but are more unique:
Gold Panning - now this one is a little tricky. It fits everything you wish for in terms of description but with the  dramas you named as example this one is not working so well. It is a gem if you look for different stories than what you normally see though.
Somebody - this is arthouse. Pure arthouse. so very explicit scenes. But the male lead is very cold. heck he is a psychopath. It would fit with "Love me if you dare" but where "Love me if you dare" would seem more harmless.