I have actually done it, although I didn't know I was trying to do it. I had flat feet my whole life as well and my family has notoriously bad feet. Mine used to hurt after standing for any length of time too. I would wear shoes constantly from the time I woke up until I went to bed. I ran in the thickest bounciest shoes I could find and my feet used to hurt. I used heel cups, insoles, arch supports, whatever, with no luck. My feet looked like a hobbits, fat and flat.
A couple years ago when I first started crossfitting I got some VFF's and started using them for a few wods, some walking, and some light jogging. I also started to go barefoot around the house instead of wearing shoes.
Then about a year ago I also bought some dharma's so that I could go minimalist at work instead of just at home. I wear those to work every day now and on my 1/2 mile to and from work to the car.
One day my wife and I just noticed that my feet looked way better. And now they even have an arch. I weigh the same now as I did, btw.
I also deadlift barefoot. I've noticed that my feet were sometimes very "tired", sore even, after deadlifts and squats. So perhaps lifting heavy has also helped strengthen them.
Anyway, yeah it works. Your feet have muscles and can be strengthened just like other muscles.
After reading this I think it becomes fairly obvious why strengthening these muscles helps.