My experience with potatoes is from growing them in the far north (alaska), where we'd plant them as early as possible (late May) and dig them once hard frost killed the plants (early September). We always wanted and needed more days in the ground than we got.
But flowers? Our rule of thumb was to look for those by the 4th of July, after about 5 weeks of growing. So as you can see they aren't any kind of indicator that it's time to dig. You *might* have some small tasty "new potatoes" under those plants but nothing at all like their final mature crop yield.