I have much the same problem in some areas. I'd say to get concentrated weedkiller and attack the bottle to the hose and spray.

Tilling any area can bring up more "weed" seeds that wil llater germinate than any other method of preparing that area for planting. Mulching will, if it is thick enough, kill the "weeds" that are currently growing and prevent ofthers from germinating, which is why newspaper is often recommended under mulches. Newspaper blocks plants access to light and without light plants cannot grow. Mulch can do the same thing, but it needs to be much thicker, 6 to 8 inches to do the same thing, keep light from those plants.
There is no real good reason to use any "weed" killer products, ever. About all they do is contribute to the pollution of our world.

I am slowly reclaiming large slopes and banks for wildflower and clover plantings.
Thie past season I had the areas weedwacked, applies round up, wait till the second crop of weeds emerged, mainly broadleaf weeds and applied a broadleaf herbicide. This spring there were few weeds there and I will be broadcasting annual wildflower with dutch white clover, in the fall I will over sow with perennial wildflower seed.
I'll be spraying one more application of round up before the clover and wildflowers emerge.
Last year I did the same on a smaller slope with great results so I am going for a larger area.
But, if you don't want to wait for a year, and want to plant a groundcover that smothers all weeds use crown fetch, it grows up a bank not down if you are reclaiming a bank.
If you want to go organic you can use pickling vinegar for weed killer, but the soil must warm up first and make sure it is applied on a hot day.
Another way would be to cover crop the area to smother weeds, Annual rye grass is a great throw and grow seed for weed control.
Hope that helps

I too abhor herbicides. I have successfully beat back the weeds in a similar situation (large neglected mixed vegetable & flower garden full of weeds & in need of organic matter) with a rotation of black plastic mulch, green manure cover crops, & fire.

Cover crop: Planted either in summer or winter, a good cover crop will suppress a lot of the weeds. However when you turn it under, there will be long-lived weed seeds in the soil which will often come back to plague you. Therefore, I follow it up with ...

Black plastic. Till under the cover crop (and/or other organic stuff), then drape the area with 4-mil or 6-mil black visqueen in 20' wide strips, weighted down at the edges with stones. This will kill every weed under the plastic in about 3 months in summer, or it can be left on all winter. By the time you pick up the plastic the organics will be totally absorbed into the soil. I generally poke holes in the plastic on 6' centers & plant squash seeds, which will love the heat & cover up the ugly plastic with luscious greenery. Sheets of cardboard (like flattened appliance boxes)also work quite well, although in my experience they won't kill perennial-rooted quack-grass as well as the black-mulch will.

Fire: a propane-fired flame thrower will also quickly kill any weeds that have sprouted after tilling. You don't have to burn them up, the heat will make the cells "explode" & the weed plants will wither & die within a few days. One caveat: Applying fire to thistle seeds makes them germinate.

Good luck.

As a follow-up to my last posting, I should say that the problem I have NOT solved is, how to prevent the Quackgrass from re-infesting the garden from outside the fence. It crawls under my barriers, grows tall up through the fence where I can't mow ... I have pretty much eliminated the Quackgrass from the middle of the garden, but the perimeter is a continual battle. Ideas welcome!

If I didn't have quackgrass, gardening would be a total joy. I've been battling it and learning about it for the last 6 years, losing nearly every battle.

Like you, I have eliminated Quackgrass from the middle of one of my gardens. When it comes up somewhere, and it does, I dig up the area, other plants and all, and get as much of the root out of there as I can. Then I monitor it and keep getting it out. In this garden, it's definitely do-able because it's not totally infested everywhere. To keep it from entering my garden, I avoided planting any of my perennials within about 14" of the garden's edge...because most of the quackgrass enters from my quack-infested lawn. In that 14" strip between my garden and the lawn, I put down a thick cloth barrier and then covered it up with the same mulch that covers my garden, so it looks like part of my garden and creates a neat "edge", albeit a bit one. Now, I pretty much just have to "guard the border". I have found that the darn quackgrass comes up right through the cloth, so eventually, I will replace it with 6 mil black plastic. However, this border patrol area gives me TIME to catch the invader before it gets to my precious plants. Because to me, the biggest, ugliest job is when I have to lift up plants and totally strip them bare of soil to get rid of the quackgrass roots that are mingling.
In another garden, I have so much quackgrass right now, I'm digging out everything that I want to keep and will be rounding up. THen, nothing will go back into the garden until its roots have been cleaned out.
It's a huge job. When I'm finished, I think I am going to change this garden to an annual one, as the quackgrass seems to continually cause me big problems there. If I make it a garden for annuals, I can round up the garden each year in the spring if I have to. Then plant a few weeks later.

I hate quackgrass.

I hope perhaps some idea here is helpful. I am learning to accept this weed and I do get satisfaction when I dig it out!