I HATE the thought of chemicals in my garden too!!! You have to be careful when flame weeding and not get it anywhere close to plants you want to live. It will wilt and kill them too!!! This is probably a better option to do before planting when the weeds are just starting to germinate and grow. Keep in mind the flame weeding will not kill the root of established plants. Will take multiple burns. But it is a good option but it has it's time and place. Have you researched silage tarps any? I think this is my next option to try.
Silage Tarp | Farmers Friend