From the photo's you provided it appears you have more of a soil problem than a water problem. Most states have and offer free soil testing and provide what nutrients that are needed to grow grass.
If me I would delay the French drain project and get a soil sample and check for Grubs. Your photo's do not show water etching or trenching in the barren spots just no grass. Brown spotting, Grubs or soil nutrient deficiency will give the visual appearance, depicted in the photo's provided.
If me I would delay the French drain project and get a soil sample and check for Grubs. Your photo's do not show water etching or trenching in the barren spots just no grass. Brown spotting, Grubs or soil nutrient deficiency will give the visual appearance, depicted in the photo's provided.