You sound like the fearful one to me. What's holding you back from asking her on a date every time you two meet? Is it the case that you don't like to ask a girl out unless you feel reasonably sure she'll say yes? To some girls that is a turn-off. You seem to be the one who pussyfoots around even after she gives you an opening. Maybe she prefers guys who are more decisive and show more initiative. You are kind of waiting for her to slide into a relationship with you. Try taking a risk.

Maybe you are a guy who fears rejection. Maybe you deal with that by trying to engage women in a manner that you hope will lead to the both of you mutually asking each other for more involvement. It's like you're saying, "Hey we could be buddies!" when you really want more than that. And she's saying, "Yeah sure, let's just do that!" (while she's kind of thinking: "If you want a date, have the stones to say so.")

It may be a wild guess on my part, but her enthusiasm, followed by non-response sounds to me like she is telling you something.

BTW, you are saying you're willing to date her because you don't want to see her hurt? That sounds like you see yourself as doing her a favor. Women like to feel that it is the man who sees getting the date as the favor.