The mystery is solved. It is Bindweed. Someone in my vegetable growers group recognised it. As soon as I saw this picture in Google, I remembered the pretty pink flowers in the lawn last year. Right where the Weeds are.
So I have done a fair bit of reading about bindweed. Pretty, yes...invasive and almost impossible to get rid of, yes as well. I will be chronicling my battle with this horrid weed. Pray for me.