My feet are way beyond strengthening. My problem is more hypermobility than PF (though that used to plague me, when I was younger and wearing crap shoes) and I have an awful habit of falling over if my feet are unsupported, either with arch support or a wide sole or the rigidity provided by a pair of boots.
I do have some vivobarefoot shoes that I wear when I'm out, but not walking miles on challenging ground. My left foot does get stronger for wearing them, but they have no effect on my right foot, which has been a mess since I ripped some ligaments, over a decade ago. I embarrassed myself in them in a busy market hall, though, this summer, turning a corner and toppling into a woman who really was rather startled. She probably thought I was drunk.
For people with current PF, standing barefoot or on flat soles while making dinner is going to do nothing to strengthen the ankles and will simply increase irritation and strain. Those sort of mundane situations are when support and cushioning are vital. Exercises to strengthen and stretch can be done at other times.
I can't wear fitflops, incidentally, as they put too much stress on my knees, which have a tendency to bend in odd directions. It's horses for courses with that sort of footwear.