I don't know what people with flat feet should do, but I do know that you can strengthen the muscles in your feet by doing stuff barefoot.
I live in a beach town, so it isn't unusual to see people in flip-flops or barefoot, especially during the summer. I've been going barefoot as often as I can and trying to get zero-drop flexible shoes except when I'm playing tennis.
My feet have gotten considerably stronger and I now understand that my feet do not need arch support. Shoes, to the extent that they are supportive, act like casts, which limits foot movement and allows the muscles to weaken. If you transition to barefoot, you'll feel actual muscle soreness, not injury soreness.
Going barefoot cured my PF.