Maybe you're thinking about the discussion at:
on Round-Up and other similar products.
I find a dandelion weeder ("The weed stick" is what I call it) to be excellent in removing tap-rooted weeds. The shallower rooted ones I grab and twist until they give up and come out of the ground. Bermuda grass I dig up and screen for rhizomes. They do not get composted and they do not get returned to the earth. They are toast.
Persistence is what it takes, especially when the growing conditions are so perfect for native plants--which is what most weeds are. So long as you remove them before they reproduce (set seed heads or send out runners) they will diminish over time.
I went to high school in Tampa, and my yard job was to keep Dad's bermuda-grass lawn FREE OF RUNNERS. He liked the blades but not the runners.
So, when I was faced with Bermuda grass in my house in Berkeley, I didn't hesitate. I dug it up and screened it through 1/2-inch hardware cloth into a wheelbarrow. Whatever the screen caught was sent to municipal greenwaste; the dirt minus the roots went back into the ground.
And, after rain is the easiest time to weed. The roots come out of the earth much more easily then than they do later, when the soil has dried and compacted around the roots.
Good luck. Try not to poison the plants again. These toxins kill everything in their path. And they kill any insects so unfortunate as to feed on the affected plants: bees, hummingbirds, butterflies, and so on. The chemicals can cut a wide swath through the ecosystem you've shown us pictures of.
Let us know how the battles go.
USDA Zone 9, Sunset Zone 17