In my opinion the root cause is pelvis tilted too much forward.

This in turn rotates both legs inwards (as in kneecaps get closer, hamstrings get farther, or toes get closer heels get farther). This brings the toes closer, so the base is narrower when walking, which worsens the balance capability during walking achieved by the feet normally not going parallel but slightly outwards, so body tries to compensate by flattening the foot to make the toes reach more outwards. Or it may also be caused simply by lack of strength in feet muscles (though I think this is usually a result, not the root cause).

My recipe is doing thorough core work to get enough muscle tone to stabilize pelvis (crotch goes up, tailbone goes down) without effort.

Only after this will the feet get unlocked to be worked on to bring back into shape.

ceejay,

See as above first.

Overloading any muscle group will only break them down, which prevents improvement. So what I would suggest to you - however much you might hate it - is to forego any intense exercise that overloads your foot muscles, and then start concentrating on strengthening the feet.

Stop running, or skipping when you feel your sole muscles no longer can support you - or better yet, start with a lower intensity, and consciously focus on working your sole muscles. But don't overload them even that way, only go until they feel pleasantly exercised, and then let them rest and recover.

Try to focus multiple times a day on keeping the arch of your foot. When home, move your foot bones around by your muscles and with your hand to feel out what motion works which muscles, which muscles work what motion.

Try to feel out how your foot feels when it has the proper arch.

Massage your foot around with thumb and fingers and try to feel out and remember the muscles. Then try to use the muscles to put the foot into the proper shape in the air on touching the floor (but no weight on it so that the muscles are not overpowered). Then put the foot on the floor while standing on the other foot or sitting, and gradually load it with your weight to feel out how the muscles work and react, and what is weak.

Then finally find exercises which work your foot muscles and which you can do comfortably barefeet. Can be as simple as standing on the ground and then deliberately and slowly lifting up onto heels, and back again - this is normally a calf exercise, but this time don't focus on your calves, but on your sole muscles. Maybe you'll need to do it on one foot first while the other supports your weight (then swap), or during sitting, so that your sole muscles are not overpowered and can keep the proper arch.

For specific exercises look up foot exercises on the net, and get into the habit of doing some regularly until you are convinced your feet can withstand everyday life and training.