Am I doing the right thing with her?

There's this lady I started talking to on Facebook Dating in December last year. She says she's been in bad relationships and doesn't want to date right now. I met her once in person for Starbucks around the time we started talking, but it wasn't a date. She's not a quick responder, as she has kids and works, so it would normally be one response from her every day or few days. After a little bit, I was fine with her having my number and texting, but she didn't want that, and Facebook Dating is the only way we communicate now. She has told me before not to let her get in my way of finding someone to date and to keep my options open. I have done that, but after some people on there tried to scam me, I only use Facebook Dating to talk to this lady and I don't use any other dating apps at all anymore.

Anyway, I have tried to make other plans with her; not necessarily as dates, but just to see her again, but she always says she can't, and she has yet to initiate any kind of plan with me. There was one time a few months ago I tried and she initially said it was too cold, even though it wasn't that bad and I would've been doing the long driving, but then she told me it was something with one of her kids and an ex. I believe in 50/50 communication between either friends or a relationship, but it's possible she may not. Gender roles may lead her into not initiating plans. She is dealing with a custody case for both of her daughters with previous exes, and she has told me before it normally takes a long time before she meets anyone off FB dating in person. One female friend of mine believes, which I agree with, that I should keep talking to her, but still be open to meeting and dating other women. However, I decided to stop asking this one lady to hangout and it feels like she views me as more of a penpal. I'll talk with her, but I believe it's too one-sided and I'm sick of being the only one trying. Am I doing the right thing with this?

Updates
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I believe she views me as a penpal and doesn't want to see me in person again, and I won't be the one initating the plans anymore. It's up to her to prove me wrong. This is why I ask this question. Is this the right approach?
Am I doing the right thing with her?
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