There is no need to dig out any of that organic matter that is ther now, just cover the area with newspaper, or cardboard, and cover that with some mulch material to hold it in place. What might then grow there will not because they will be deprived of access to sunlight that all plants need to grow.

If the above ground growth still has weed seeds in it, you can cut those off and dispose in the trash to avoid putting any more weed seed in the ground. Otherwise, a lawnmower will make trimming the above ground parts of the weeds down to size and mulch them for the soil. Assuming these weeds are annuals, their roots will be dead and should till into the soil fairly easily.

If there are no seeds being produced now there is no good reason to remove any above ground growth of these "weeds" and throw away the valuable nutrients those plants could contibute to your soil. In many forward thinking places putting yard waste inot the trash is prohibited anyway with the idea that people shoiuld recycle that waste and not put it into a landfill.
The only yard waste that one should not recycle on site might be diseased plant material or something like Poison Ivy, Oak, or Sumac.