The girl you are interested in sounds just like me when it comes to the opposite sex!
From my point of view; you will reach this stage with her that you're talking about but things will just take longer than normal and you'll need to put in a lot more effort. Girls like us often become shy or enclosed when it comes to males/relationships, especially if we're not used to the attention! We'll flirt but nine times out of then, we like the men to put in all the work. It's just how we are and we like to know we're making them right decision when agreeing to date someone... because quite big commitments come with it.
I'd just ask her out for a casual meet up - for me, that's less scary than an actual date! If this girl really likes you, I'm sure once that front barrier has been broken, things in your relationship will progress. But don't be too forward and in her face, play it cool and friend like!
I hope this helped!
Natalie xo
Be the change you wish to see in the world - Gandhi