I'm dealing with a similar situation with a member of the morning glory family that is related to the bindweed, and have similar growth patterns.
I never heard of bindweed before reading your post, and did a quick internet search and found an article that made a lot of sense to me.
It explains their roots spread far from the plants and that physical removal might not always work. If a small portion of the root is left in the ground new plants will grow. I've experienced this and it seems the more plants I take out, the more grow back more vigorously.
I'm going to try the two suggestions made in the article:
1) pour hot water about two to three feet from where the plant is growing while covering the area to the plant stem. This will kill the roots and my guess is it might have to be done a few times. Also, my guess is that boiling water should be used.
2) It is also suggests that constant pruning to the ground will kill the plant's roots as they constantly use all the energy it takes in and kills them.
Don't know if these techniques are effective but I'm going to try them. Everything else I've tried has not worked and my feeling is I'll be able to spend less time working with these weeds and more time with plants I enjoy growing.
Hope this helps.
