Right, this is harsh but what I think you need to hear:
I think you should stand in front of the mirror and repeat the following to yourself several times a day:
"The children come first. It's not about me."
See if it sinks in.
This isn't about what's best or right for you, for your girlfriend or for the children's father.
What's best for the children is having a calm, stable environment, their mother facilitating contact with their father and their father being happy to see them when he does. Would it be better for them if he saw them regularly and joyfully and paid towards their upkeep? Why yes, it would. Does that sound like a likely scenario without upsetting aforementioned calmness and stability? Why no, it doesn't.
Even if she could somehow magically force regular contact, I can't imagine she'd want to send her children off to someone who resented having them and therefore probably wouldn't care for them properly. The well-being of her children is more important than her having free time. When he is willing to have them, she drops everything because she wants her children to be able to take advantage of seeing their father when they can and their well-being is, again, more important than her free time. Is it fair? No. But it sounds like she is selflessly doing what she feels is right for her children.
Leave it alone. Your girlfriend is doing the best she can for her children based on the information she has, which - believe it or not - is likely to be a lot more than you.
Stop beating your chest and decide whether you can swallow your pride and sense of injustice for the sake of the children, which is what it sounds like your girlfriend is doing very well, god bless her.
If you can't and you stick around anyway, you're going to be patting yourself on the back for being a 'good role model' while actually making the situation worse for them. It would be better in that case if you just leave now before the resentment that's clearly simmering bubbles over.