Of all the pest problems, weeds are the most difficult to control organically as far as I can tell.

In my experience there are really only 2 options. 1) Systemic herbicides (and there are no organic systemic herbicides). 2) Determination and persistence.

What I have done, and my garden is small so I can afford to use time-consuming methods, is mostly to scrape off the top two inches of lawn/ soil and pile that up elsewhere in a spot where those sorts of weeds don't matter, and then sheet mulch over the remaining dirt. This does not lead to a weed-free area, but it really knocks down the population. The other thing that helps is to plant taller and denser vergetables such as broad beans, tomatoes or potatoes. If you get them growing densely enough for long periods, they will further reduce the population of weeds by shading them.

If you can get old wool carpet, that is more effective in sheet mulch than newspaper or cardboard.

For the weeds I have, the method I have used is ok. I still get couch grass/ twitch coming up. I mostly just pull it where it comes up. If I find there is quite a bit in one place, I might pull up the mulch and dig out as much of the roots as I can.

I don't have kikuyu but know it from when I lived in a warmer climate. I think that it mostly resprouts from creeping stems above the ground, so scraping off the top might work for that. Bindweed is like couch grass - long, buried rhizomes. It will definitely keep coming up even if you scrape the top, but it will be reduced. Crabgrass - if we are both talking about the same plant - should be like kikuyu, mostly sprouting from above-ground parts.

I've used sheet mulching in a number of different gardens and regions of NZ, and I've found that success depends on a) what weeds you have and b) how thick and difficult-to-grow-through you can make it. The method I used isn't from the permaculture manuals, but it is what i have come to after years of trial and error (with plenty of emphasis on the error).

I don't regret removing the weeds from my vegetable garden first, it has made life a lot easier. I've used the dug out soil and weeds elsewhere, and although I took some of the fertility out of the garden when I took the weeds away, I replaced it with lots of manure, compost and other random organic matter. I just lay this on the dirt and covered with newspaper etc, and the worms very kindly dug it all in for me. I ended up using much less in terms of off-site inputs to my garden than if I had used a thicker layer of sheet mulch, so I think the method I used was fairly sustainable, except that it was hard and I would have not been able to handle a larger area.

I've attached a link to my blog - it does cover some of what I've mentioned in this post. Hope this is useful - I know I'm in a completely different climate etc but you might find something helpful in my experience.

Here is a link that might be useful: Belladonna Bess garden blog