I guess I have a little problem with your situation because I feel that you have this perfect sex scenario in your head, and that if anything doesn't follow that scenario perfectly, then the sex is ruined. Don't get me wrong, I totally understand your frustration, but at the same time...

My fiance and I are a heterosexual couple, and I sometimes have the same problem with him that you have with your girlfriend (it has nothing to do with gender). He really doesn't like initiating either. And frequently, we'll initiate in different ways. What I see as starting something doesn't always come across as starting something to him. This can lead to us getting frustrated with the other because we'll both be like "Didn't you get the signal I wanted to fool around tonight?" It's especially frustrating when we both want to, but our signals just didn't make sense to the other person, and we both missed it. So, yeah, I totally understand where you're coming from. I also get the whole feeling less desired, less wanted that goes with someone who doesn't like to initiate (It doesn't help that I have a higher sex drive than him; he also puts more importance on cuddling and simply being close. Seriously, if he got his cuddles every night, he'd probably be fine with sex once every two weeks).

But I've come to realize that if I just hold out on him, waiting for him to make a move, I'm essentially shooting myself in the foot. Now, if I really want some, I know I have to go after it myself and make things painfully obvious, aka, I just get in bed naked. It works! Another thing I do, which took some time for me to be okay with the idea, is accepting the fact that I need to be obvious, and say "Hey, wanna make out?" or "Sex tonight?" I know it takes the "romance" out of it, but it's not unnatural. I also don't think it ruins the spontaneity. You're spontaneously asking for it, and it can still be passionate. What's better is you don't set yourself up for painful disappointment later.

I'm glad that you were able to talk to your girlfriend. Perhaps you could suggest going to the gym together or just going out for exercise together (walks, bike rides, hit a tennis ball back and forth). The more you fight about it or rant/rave/complain about it, the less often it'll happen (trust me, I've been there). I think you both need to be open to change in this situation. She has to push herself to initiate more (or initiate in a way you understand), but you also have to be willing to pull away from your perfect scenario and blatantly ask for what you want.

I think you two can move past this, you both just have to be willing to bend a little and change up the roles you're comfortable with. I've personally been meaning to check out the book, The Five Love Languages, or whatever it's called, because I feel like this boils down to a different way of expressing love. I haven't done so yet, but it's on my to do list.