Tilling constantly for the several years required will kill any living plant and in the process destroy the microbial life and structure of the soil. It will also make the original poster's yard into a desert, which would be highly problematic, one might think. As I have gardened and farmed organically for the past 40 years and battled bindweed for 6 years in an Adirondack garden, I know (not imagine) that repeated cultivation and pulling hardly eradicates the stuff, only beats it back for a while. I did this for years to keep it in check in a quarter acre garden and the stuff returned as soon as I moved. Of course, it grew in all the neighboring properties so it was only a matter of time.
Birds, bits of roots on a tiller blade, wind and any piece of root a couple inches long that isn't immediately brought to the surface and completely dried produces a new plant. Those roots 20' down are difficult to kill and impossible to pull up. The seeds can remain viable in the soil for up to 50 years. Imagining it away doesn't work.
ATTRA, Organic Gardening Magazine and many others writing for the organic & sustainable viewpoint, describe the horrors of this plant and the difficulty of controlling it. They are not controlled by the chemical companies. Perhaps they just lack your imagination.
Wayne