I'm in the same camp as eeyoretigger, because when we took on a new and very very weedy plot last year it as obvious we could never get it all under control at once. Covering a large section with landscape membrane, well weighted down so it stayed in place meant that all the weeds were totally dead a year later (We don't have any bindweed, which takes longer)
Those weeds also didn't set seeds to infest other areas either, which was good.
Even if you cover most of your plot, it'll not harm it at all and allow you to focus on enjoying growing something to eat, and getting your little ones to enjoy being there too