My word, you HAVE got a problem there, Steve. If you can clear all the rubbish and dig the place over, wait for the bindweed, etc. to get growing really strongly, and THEN use a systemic weedkiller, possibly using it at double strength, and giving the whole area a season without planting at all, you just might be able to clear it. You will probably have to repeat this a few times during the season. By putting a membrane over the area immediately after treatment, you will cancel out it's effectiveness. I have a similar problem with bindweed, couch grass and mares' tails in my garden, and this is how the local farmers' advice service suggested I proceeded. I think you should give it a go before resorting to staging and pots.
If you do decide that pots are the way forward, though, carrots can do well in those pots, and salad crops would probably be ok, especially spring onions. You could try a few spinach plants, or even a few dwarf french beans, too. Certainly herbs would do well.
The very best of luck, Steve, I think you have quite a job on your hands. Let us know how you get on.

I agree with Gattina..leave it for a while after treating.We had lots of Bindweed too,but as it had been newly landscaped,by the builders,and planted up,I wasn't able to treat it like you will be able to..but we did finally get rid of it...mostly by hand..it took a while,but eventually we triumphed...you have the ideal opportunity to start from scratch good luck..It will be worth it in the end..