They are right, burning is not an answer to weeds. Weeds are the plants that hang on on the edge, naturally they grow places like on the edge of rock formations where there is not enough soil for trees to smother them, then take advantage of forest fires for a bit until the trees come back. In nature some are quite rare plants, but we create conditions where they can thrive by knocking out the competition.
Even if you did burn and it worked, which it won't, you would have to till before re-seeding with grass. So why not till, water to germinate all the weed seeds, then till again (It is a lot easier the second time) to turn them into green manure for your grass before they flower and make a new lot of seed. There might be some gross roots you need to pick out, but it will work better than burning.
On the other hand I once got a piece of land like that and covered it in old carpet, then I cut slots in it and slipped a potato into each. The carpet smothered the weeds and the potatoes outgrew the few that found the slots. The potatoes grow across the surface, you can lift the carpet and pick the spuds up where they are half buried. You won't get a great crop, but it is something and a few months smothered will kill nearly all weeds, they will be rotting and ready to till in. Cotton or wool carpets will rot and you can till them in too, artificial fibres you will have to lift. Heavy duty black polythene can do the same job, but old carpet comes free and doesn't add any more plastic to the world.