I decided to look up google images of burdock and Danz, you were right. In fact, after seeing pictures, I'm now thinking that third picture with the broader leaves might actually be burdock. So that made me curious about what the plant is that I've called burdock and from what I can find, it is related - curly dock (even though the leaves are not what I would call curly). So I bet the same thing applies that the root is edible.
Danz, I was also giving some thought to your issue with the area you mentioned that doesn't grow anything but weeds and I'm wondering if you aren't caught in a cycle of tilling and spraying that is preventing the area from becoming fertile. By spraying round-up, you kill the weeds but enough remains in the soil to make it difficult for things that are more susceptible to round-up to grow. The things that are the least susceptible to it eventually regrow, so you are caught in a cycle of spraying and infertility, with nothing but weeds growing there at all.
I truly believe that ANY patch of ground can be "reclaimed" and made fertile, whether it is sandy, clay, weedy or whatever. All it needs is a LOT of organic additions and no more poison. There is a book called "Gardening Without Work" written by a lady called Ruth Stout several decades ago. Although it is quite an old book, most of the advice in it is still relevant today and even better, she is a delightful writer with a great sense of humor so it is an easy read. I thoroughly enjoyed it when I read it years ago. Now, I would say the "without work" part is a bit of a misnomer as I follow her methods and still do a lot of "work". However that is primarily due to the source of organic material she was using compared to me. She didn't raise livestock so she bought in old hay that was delivered to her and all she had to do was spread it on her garden area in a thick layer of mulch. However I don't buy mulch, so have to gather it from the chicken coop, sheep barn and so on. The gathering is definitely work - the spreading is pretty easy.
Her basic principle is: spread a thick layer of mulch - at least 8" - over the entire plot. As it breaks down and decomposes, it adds nutrients and fertility to the ground below. And, the thick layer prevents weed seeds from germinating so you will not have to pull weeds. By laying out garden beds and walkways between, you can avoid walking on the garden beds so the soil does not get compacted and therefore does not need to be tilled. I was actually using her methods (or similar) before I ever even heard of her book so while reading it I was just saying "yes, yes, yes" to myself the whole time. It really does work. We had heavy clay here but every year the soil I am planting in is richer, more fertile and easier to work by hand. We don't till but just keep adding mulch as the layer breaks down and gets thin. I didn't weed at all last year. Oh sure, a few things will occasionally try to grow through the mulch but I just pull the individual weeds (which is easy because the soil is so loose and friable) and then throw them on top of the mulch so they become part of it. I also didn't water last year. The mulch layer retains moisture well enough that it wasn't necessary. Planting is pretty easy - pull back the mulch, sow the seeds, and once they've germinated, snug the mulch back around the stems of the seedlings so they don't dry out too much.
What I would do in that patch you have if it were mine is to stop tilling and using poison, and instead pull the weeds you have (yes, it takes time initially but is more effective in the long run). Then I'd cover the area with 8" or more of mulch and leave it to sit until next year. You'd have to keep adding mulch as the initial layer breaks down, trying to keep it at 8" or more at all times. By next year you'd be able to plant into that spot with very few weed issues. Continue to pull any weeds that do try to come up and there really will be fewer and fewer over time.