Flagwaver- so am I! But I'm also a certified Pesticide Applicator, and I spent several hours twice this spring running around with a backpack sprayer full of RoundUp and spraying too many burdock and yellow dock plants!
Sometimes it's the only way you can deal with them
Dock is especially nasty mostly because it seems to be able to set seeds after you've tilled it under!
If it's starting to put seed on- AT ALL- your best bet is to somehow (know someone with a sickle bar mower?) cut it off and rake it up and either dump it in the trash or burn it- WELL.
If it is nowhere near the "setting seed" stage (no tassle starting to come out the top), a good spraying of Roundup will kill it off permanently.
RoundUp is probably THE least toxic spray I've used at any time in 20 years of being licensed. We actually do farm nearly organically, and I garden the same way, but some things just don't work that way. I've tried all the "boiling vinegar spray" tricks to get rid of both burdock and dock- and they've just continued to multiply.
RoundUp, unlike most herbicides, actually does what you said "absorbs the poison in the plant, sends it to the root"- and is immediately deactivated by contact with soil.
Now, given that it's in a planting you want to keep, you can't just spray it. But you CAN *carefully* paint it on the dock plants with a roller or paintbrush- and as long as you get it on at least 3-4 leaves on each plant, they're goners...
Just don't trip and fall and spill your little can of RoundUp- because ANY plant it covers or splashes heavily on is in trouble. Some folks DO spray plants that are in the midst (or close to) valuable plantings, but they hold cardboard or something as a shield.
And if you have a terrible accident (and splash the stuff all over a plant you love), you simply grab a hose FAST and wash it off well. It sets and is impervious to rain within 2 hours, but you do have a short window to rectify mistakes.
Summerthyme