Not to that extreme.

What I am suggesting is as follows:

suppose that mental health were taken seriously in Korea (counterfactually, according to OP).

What would you expect from a doctor/nurse/therapist who takes MH seriously?

You can expect medications from a doctor/nurse but not a therapist.

You can also expect a doctor/nurse/therapist to advise your gf to use non-pharmacological tools to get out of her depression (exercise, sun exposure, etc.)

Since it is not rocket science to - exercise, walk in the sun, etc. - and the advice of a (non-existent in Korea) doctor/nurse/ therapist to use those tools can be predicted, you might as well suggest them to her. You cannot prescribe medications but you can tell her to go outside and spend 20-30 minutes in the sun.

And if that fails, I think there is nothing you can do - she appears resigned to leading a non-life as a hermit who plays video games.

PS Since she responds to your sexual overtures by citing her being tired, you can try:

1) make those overtures first thing in the morning rather than at bedtime, and/or
2) explain to her that all you want (I assume this is true) is basically advanced hugging - you do not expert cardio marathons from her, nor do you expect sex in unusual, contortionist positions, nor will you spank her tush to the point of her being unable to sit on the tush and play her video games (mentioning that since playing video games seems to be one of the only two things that she likes - that, and extreme couponing). So... what is that big of deal? I just do not get it. I mean, there is an undeniable gender difference that will not go away with technological progress - men do need an erection/arousal for intercourse and women do not. So, OK, she is not aroused, that is fine. But, it is not that she is actively repulsed by you - right? If she cares enough to kiss you in a non-sexual fashion, that means that she is not repulsed by you. So... what is such a big deal with letting you hug her a bit more, with a few thrusts added for good measure? Or, would you not engage in intercourse unless she is actively passionate with you? What exactly is the problem?

Unless I am missing something obvious... You are her long-term boyfriend and she wants to live with you and wants it badly. So what is the big deal? If she allows you some advanced hugging with a bit of thrusting, the experience will be neutral for her - NOT NEGATIVE, but neutral. So there will be what is called a "Pareto improvement" in Economics - you will be better off and she will be no worse off, and overall there will be an improvement. Why would not she do something that will make you better off without making her worse off is beyond me. She has not been abused nor raped so it is not as if she were triggered somehow.

I have a friend whose wife did not want to have sex for a year and half after the birth of their only son. He complained to me a lot, over email. It seemed utterly ridiculous to me but at least his wife had a POINT. She was, indeed, too tired (she was working full time and he was a WAHD - work-at-home-dad), so she did not want sex because for her sex had to be earth-shattering. Earth-shattering or none at all. To have an earth-shattering experience, she had to be well rested. It seemed a little non-generous and a little selfish to me - he would have liked a quickie without high expectations - but she insisted on "Earth-shattering or none at all". In other words, she was a perfectionist to an extreme degree in the realm of human sexuality. I think the point was weird, but at least VALID. And your gf does not seem to have any point at all.

I am sorry this seems to lead to a dead-end situation for you...