I don't doubt that the seed is as you say, it's a wild / feral form of amaranth, and I wonder if native Americans didn't cultivate pigweed in times past... The trouble is pigweed sheds the seed very easily, I would opt for amaranth.
However, pigweed is a first rate vegetable, (prized in some parts of Mexico). Pick the soft tops from plants before they produce much of a flowering head, they aren't fibrous, and very nicely flavoured, amenable to blanching steaming, or cooking in the microwave.
So although it is a nuisance weed, I am pleased to have it appear, it gives weeding a whole different complexion.