Trying to think back to most of the women I've slept with....... I don't really think I have ever turned medium interest into high interest. However, as my game has increased, the hotness level of girls I sleep with and date now is higher than it used to be. Game can't be discounted because it is critically important in the first few minutes of the interaction. Once you understand good game, you are not going to turn yourself into a game EXPERT and enjoy more success with women. You have to be confident, have strong eye contact, ability to communicate intent, entitlement, dominance, awareness, sexuality, physical escalation, escalation in general, not take women too seriously, and know how to "handle" women's actions. Once you have that, the rest, as OP stated, is entirely about the girl and what does it for her specifically. If that's not "you", game will not help.
Another way to put this is that, for every girl who's "type" you are a match for, game ensures that you hook up with 100% of them, instead of a lower percentage. If you aren't her type, no amount of game in the world is going to get you in there.
Another thing about game is that it increases the number of women you interact with.... guys with game will generally know how to put themselves in places where there are a large number of attractive women, and guys with game will also pull the trigger and initiate contact with women, or at least notice when a woman is trying to give him signals.