The friendship route can work, but yes, it can be tricky. The reason being the she simply wants you as a friend with no strings attached. I gathered you finally stopped with the begging, pleading, and pushing when you saw it didn't work and now hope to win her back by being her friend? The problem is that you want to be more than her friend, and you are already wondering when you can start "pushing" for more.
If won't go well if you do. If it doesn't run her off, it will put up her guard much higher and she will feel that being friends is not what you had in mind at all. And she would be right.
The best a LBS can do in the friends stitch is to hope she will fall in love with you all over again. But it will have to be without you planning and plotting for when you can expect more from her. When she says "just friends" to the man she was going to D, that is exactly what she means. Friends with no benefits!
It is not about what you feel should work in your M. It is about doing the work that gets the right results. Do what works!