Cycling and walking/running are two different beasts.

On foot, your foot is in contact with the ground, and we have a large number of bones and muscles that have particular purposes, including supporting and flexing the arch, catching the body and propelling it forward as you take each step. If your foot muscles get sore when you're on foot, that's a sign that they need to get stronger - just like sore muscles in any other part of your body. The more flexible the soles of your shoes, the better your feet will be able to do their job.

On the bicycle, your feet are transmitting power through a small contact patch between foot and pedal. The size, shape and location of that patch doesn't change from pedal stroke to pedal stroke. When the sole of a cycling shoe flexes, and your arch flexes during the pedal stroke, no purpose is served - your heels and toes go down into empty space, and your foot muscles are stuck scrambling to hold your feet stable in a position they didn't evolve to work repetitively. The stiffer the soles of your cycling shoes, the better your legs will be able to do their job.

Think of holding a plank, because it's really similar. Your spine and core muscles evolved to hold your body upright. Planks are good exercise, because they're asking those muscles to work against gravity in a different direction than they're used to, and that'll help you be stable in motion, when you're not perfectly upright. But could you hold a plank for five minutes... ten... an hour? That's basically what you're asking your foot to do if you bicycle in flexible shoes.