Do note what the article says - the cut material is controlled waste and cannot easily be disposed of and poses a significant risk in further spreading the plant.

You may also have only fought it back to dormancy. The reason glyphosate is recommended is that it is transported by the plant to growth centres where it interferes with normal plant cell operation, and so can kill the plant deep down. Even then, it can take many applications to be fully effective, and needs to be applied at the right times in the season. It is also normal that it is not immediately effective, so patience is required.

Now, whether glyphosate is safe for humans.... they use tons of the stuff in America, liberally spraying "roundup-ready" GM food crops. I don't think that's necessarily good, but if there were a problem, I'm sure the effects would be seen now, and there are a number of studies with conflicting and inconclusive results.

That said, I don't really like herbicides and pesticides and use them rarely and cautiously, but Japanese Knotweed does demand tough action to control.