I abhor grocery store peaches, as most are picked waaayyy too early. Those, however, sound quite nice.
Even if ripened at home, they don't create more sugar after they are picked, but acids break down. That's why every where I've lived, I plant peach trees. I my wife just ate the last off my early peaches, picked only when completely soft to the touch. My late tree won't be ready for another couple of weeks. After next week, it will be war with the raccoons and squirrels until time to pick. An electric fence surrounds the tree two inches off the ground.