People worry about horses having flat feet, much more I think than the horses ever do. Part of the problem, is that sometimes flat feet are lumped together with thin soles.
A 'flat' foot, with a good solid, thick sole and no underlying inflammation should be just fine. You often see this conformation with some of the heavy horses and cobs, and they are often some of the most successful barefooters.
But a horse with a thin sole is going to struggle. The good news is that you can fix this; if you follow the DETEC3T protocol.
The two feet profiled above both started with flat feet and thin soles. The first foot belongs to a horse with extreme metabolic issues, so tends to only be 'rock-crunching' when conditions are optimal. Winter, no grass, thoroughly soaked hay, etc. regardless of how thick her soles are. But the three pictures do show that a flat foot can gain concavity.
The second foot had one of the thinnest soles I have come across. Taking the shoes off allowed better circulation to the foot and a natural foot form to develop. As you can see from the last three pictures, this foot has developed some concavity (and a thicker sole) relatively quickly. The horse can now hack out on grass and concrete without boots. Stones are still too much, but at the current rate of progress, it won't be too long.
So what is DETEC3T (long term readers will know this already so apologies)
Diet - feed a diet appropriate for your horse, which means knowing if they are 'metabolic' or not.
Exercise - feet grow in response to movement - so get your horse moving as much as they can, comfortably
Time - give the horse time to adapt, to any change in diet or exercise, to grow new feet
Environment - a hoof responds to the surface it lives and works on, so vary the terrain
C3 - commit to growing a healthy hoof, be consistent with diet/exercise, commuciate - listen to your horse
Trim - learn what is a good trim, get one when the hoof needs it, don't try to cheat nature