Personally, I get on my knees, pull up what I can by hand and put these in the compost bin. I take a trowel to the dandelions and weeds with a taproots - these go in the council compost bin. I then cover with as much compost as I have available normally 2 inches, more if I’m planning to grow spuds or something hungry. Then leave the worms to do the work. Come spring you may get the odd seed germinate on the compost but the top will be light and the weeds will be easy to pull up.
I am not a fan of plastic membranes.