Do I stand a chance to win her back ?

My gf of 4 years and I broke up somewhat mutually and very amicably.. I thought it was what I wanted.We had communication issues towards the end that was driving us apart and stoping any love from nurturing. After the break up I felt fine no hard feelings towards her and no desire to contact her. On about day 5, I reached out to see how she was doing and to let her know I accepted the break up and believed it would be the only way we could remain friends. We had a great talk, we connected like we hadn't in months. That left me wanting more. I realized I had made a huge mistake. When we opened up to each other we both agreed we felt so close. I asked her if she wanted to get back together on the promise that we would be more open to one another. She said she thinks we work better as friends because there is no stress to impress each other. That was very hard to hear so I made the mistake of begging which just made it hard for her, she said she needed time to move on. In the last 3 weeks my heart has not waivered. I still want her back. She is being very open and honest with me about what she has been doing with the limited contact I have initiated with her. She has been seeing someone else and appears to be having a good time. Each time we talk I feel closer to her but it always leaves me wanting more. I asked her not to completely close the door on us, that if it felt right and natural someday we would give it another go. To that she said she was open to anything but didn't want it to hold either one of us back from potential happiness elsewhere. I'm trying to take things slow and not to interfere with her and the new guy. I am just wondering what the best course of action to take here is. Ideally I'd love to have her back but I can also find the silver lining in knowing that she is happy even with it being with another man. Should I just let it be or keep slowly perusing and trying to win her back over with the changes I've made ?
Thanks in advance