You know, I am wondering if 'my' weed is Creeping Charlie. The online info refers to blue flowers in the early spring: the weed with which I am obsessed does not have flowers unless it is left to grow stems. (At least I think that's what happens.) The photos online show leaves that are more deeply scalloped. Of course I could have another variety, and clearly whatever it is it is similar to other creeping ground covers. Anyway I have selected a few areas that are relatively large and manageable to conduct some experiments as suggested: hoeing out and monitoring, covering with cardboard and monitoring. I'll let you all know how it goes after a while. I have appreciated all the information and advice from this thread.
All this to avoid a little Round-Up. I have a quart of concentrated that I am just finishing off from 2012. I use it sparingly wherever I have a problem weed that is not controllable by hand pulling. Mostly, that means those kinds of weeds that spread by deep rhizomes or spread billions of seeds. Almost all of my beds have ground-covers which are mainly weed-free because the earth is not disturbed and shielded from sunlight, the two biggest generators of weed seed germination. I have ten times the eye candy of most gardens and one one-hundreths the weeds. And, very few interlopers into my vegetable garden which is mulched with 3 to 6 inches of leaves, year-around. It's been noted above: toxic is a level, not a material. Toxic levels of manpower use will lead to giving-up, and giving-in. The weed will beat you.