Has Pink Family flowers such as those produced by Chickweed species.

Has never produced a flower that I've seen, just hoards of 'stick to everything' seed pods.

I agree with the Chickweed. Drymaria cordata - Tropical chickweed or Sticky weed.

Pull/dig it out and do not let it go to seed would be my best advice.

Thanks. Given how badly it has already spread, I think my backyard is toast...

It's an annual so should be killed off with cold weather. If not, any broadleaved weed killer will work but you will probably also want to apply a pre-emergent later in the season (closer to spring) to nip all those seeds before they sprout.

I live in Central FL so no weeds actually ever get killed off by our weather. Weed-n-feed products don't even phase this stuff - although I have yet to put down a weed treatment by itself. I know that baking soda will kill crabgrass (at least the above ground part), so I tried it on a small patch of this weed and it looks like it might do the trick. Will try some larger areas and see what happens.