Great to see you working on moving fast and getting fast escalation down. You probably could've had this girl, but for the making out. See the notes on kissing in the post on how to break the ice.
Essentially, kissing acts as a sexual tension release - once you've kissed her in public, the odds she goes home with you / sleeps with you go down dramatically. It's also somewhat of a commencement of mating, and can count as a "failed mating attempt" unless she perceives YOU as the one having broken it off. If SHE broke it off, she'll backwards rationalize it as not being very interested in you (it's an emotional thing). Once mating's failed, you quickly get written off as "not a guy I want to sleep with," and women will move it quickly to being just friends.
So, I'd be very surprised if past boyfriends are the reason why she doesn't want to date YOU. Remember that if she's dating guys who treated her like crap, they're probably also guys who don't care much about women, and probably picked her up fast by default. That's what she's used to, that's what she likes, and any guy who isn't a fast-moving asshole who treats her like crap gets written off as weak / friends material.
Best bet here if you really want her is to just tell her something like, "I like you, but I'm not really interested in being friends with you. Let's talk if you want to hang out sometime, though," which sounds asshole-ish, but a girl who dates bad guys isn't looking for a nice guy to save her... she's looking for another bad guy. I've pulled this same move on girls with similar relationship histories, and they eat it up. They whine and complain, you cut them off for a month or so, and when they come back to you suddenly they're desperate to be with you.
So, you say that, then, do cut her off, until you have another chance to move fast with her again sometime - but that time, don't kiss her, just get her alone and make things happen.
Fair warning, you probably won't want to date her though. Women who make great relationship partners tend to date MEN who make great relationship partners, whereas troubled people attract troubled partners. She'll be a handful in a relationship, and not in a good way.