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In traditional karate (not kumite competition) performing mawashi-geri (roundhouse kick) such that one strikes the target with the ball of the foot is preferred over striking with the top of the foot as in sports karate or with the lower shin as in Muay Thai or kickboxing.
A similar phenomenon seems to exist regarding ura/ushiro mawashi-geri striking with the heel versus the ball of the foot. Why is this the case and is one style definitively superior for full contact fighting?
Not sure if this truly answers your question, but I'll give an answer from a Taekwondo point of view.
In our kata (we call them poomsae, hyung, or tul), we were once taught to use ball of foot. And when we sparred, we did the same. In self-defense context, we assume we don't have complete mobility with the ankle, such as the case when wearing boots for example. So we learn in self-defense to kick with ball of foot (or even toes, depending on the rare application). In sport, we kicked with the ball of toes to get around the guard.
Nowadays, while it ultimately depends on the school, Taekwondo-in can choose ball of foot or instep as they deem. In WT poomsae competition and in Kukkiwon grading, we are given the leeway to choose, so long as we are consistent. Don't use ball of foot in one technique, then change to instep in another. In poomsae, we never kick with shin, because we never strike a target, so this is moot. But in sparring competition, kickers almost universally use the instep for two very practical reasons. In WT competition, they never use the shin, because that would be an illegal technique (only the part of the foot below the ankle is a valid striking weapon). Using the shin, if accidental, will go unnoticed. Deliberately or repeatedly using the shin or knee will garner a warning for use of an illegal technique.
As to the two reasons, the first is range: you are given a few free inches of distance. In competition, you generally need not worry about damage to the instep, because (1) the foot is protected; (2), you want to score points, not KO's; and (3) your opponent also has padded protection (but only on the hands, head, and trunk; not the legs or arms, and while these are illegal striking weapons, accidents can and do occur.)
The second reason has to do with electronic scoring - the sensors in the padding. In the "old" days (even as recently as Rio Olympics), the chest protectors had known bare spots - places where you could strike but there were no sensors, or the sensors didn't read well. Using the instep gives more surface area to contact more sensors, thereby gaining advantage when these sensors were not present or malfunctioning because there would be a chance of a nearby sensor that could pick up on the strike. The sensor thing was not a game-changer for those fighters choosing to use ball of foot; most (if not all) had been using insteps for years. But the sensor issues definitely placed more nails in the ball-of-toe's coffin.
You may note that throwing the round kick feels differently between the two options. With the ball of toes, you have less distance (the radius of the kick goes from the knee to ankle) and so you may find you turn faster. With the instep, the length of radius is slightly longer, thereby potentially slowing the kick ever so slightly. Also, the muscles needed to pull the toes back (ball-of-toes) are different than the muscles necessary to push them forward (instep), and that has effect on the rest of the leg's coordination of the kick.
I've discovered that kickers who are used to kicking with ball-of-toes tend to adapt well to instep, but the same could not be said of kickers used to instep and try to use ball of toes. This has effect on breaking. For one or two boards, I'll easily use an instep. For more than that, I'll defer to the ball-of-toe. But others cringe at breaking even the measliest 1/4" boards with insteps (despite sparring with instep) and completely bungle the break because of balance when they defer to ball of toes. I don't understand why, it's just what I observe.
I offer a guess, even if probably not related to your question (feel free to call me out on it! lol)
To use the instep, the gastrocnemius and soleus muscles (ie, the calf) must contract.
To use the ball of toes, the anterior tibialis muscle (ie, the instep) must contract.
Larger muscles tend to be more easily contracted than small muscles, so in this case, it takes more work (and skill?) to effectively use the ball of toe than it does the instep. Perhaps not surprisingly, when neither muscle is contracted, the foot will tend to flop, as anyone who watches young children try to throw a roundhouse will attest is very common. And in this case, the foot's instep will naturally open (perhaps also aided by the circular movement of the shin and centrifugal force?) the instep is the result, however unintended.
I doubt any of this was factored into the use of either method, though maybe it explains why some people choose instep over ball-of-toe when given a choice.
The usual answer to this is conditioning
The bottom of the foot/ball of the foot supports our weight all day long, it is naturally designed to take the harder impacts resulting from kicking something hard.
The instep is not designed for this impact - and an injury (bending of the ankle - simple bruising - or even a broken meta-tarsal) much easier to obtain without care/conditioning and good technique.
A couple of other considerations
The ball of the foot gives additional range over the shin as well as being a smaller point allowing for attacks to smaller areas.
"round" house kicks in many arts should actually travel in a straight line - the body rotates forcing the knee forward and then the lower leg extends. Meaning the foot travels in a straight line from floor to target (where the target would be at an angle to the attacker) - performing this version of the kick correctly means that pointing the toes will result in them striking first and pulling the toes up whilst pointing the foot (much like a front kick) strikes with the ball. With this kick neither the shin or top of the foot are really viable.
The Reverse
Pointing the toes to strike with the ball of the foot will give a bit more range - but the ball of the foot is using muscle strength to stay there and will just bend back if any serious impact is tried. The heel doesn't have this issue.
It is common to see good sparrers fail to break because they naturally point the toes for range with the reverse - giving the breaking boards a slap instead of striking with any force.
Compared to the instep, the ball of the foot can take more impact without injury (though it's easier to injure your toes by accident), and having the impact focused in a smaller area of the target means much more localised penetration and damage: you have a better chance of e.g. breaking some ribs or penetrating over the liver enough to drop an opponent. When kicking with the ball of the foot, you also reach in sideways from any block at the shin by ~20cm, which means you may still get a good hit on someone who just wraps their arms around their head.
A conditioned shin is very strong though, great for low kicks while standing around the edge of or inside punching range. Have to be very flexible and close in to kick head high on a similar-height opponent with the shin though.
Personally, for the first decade or more of my training I only kicked with the ball, but now I'm equally comfortable with shin / ball / instep. It's good to select based on the target, distances, whether you're really trying to injure someone....
For ura/ushiro - the back of the heel can be conditioned to be very, very strong, and as it's very hard the impact to the target is very sudden and destructive. Kicking with the sole of the foot gets a smidge extra reach but is far weaker. If you strike nearer the ball you've only got the calf muscle to keep the foot extended: it will produce a much softer impact spread over much more time, with far less breaking potential. If you strike near the heel anyway, your more likely to injure your ankle joint, though the bottom of the heel doesn't need the conditioning effort that the back of the heel does: we walk around on it all day anyway.
In reality, a lot of "kicks" were pushes when used within the context of original applications from which the "kick" was taken, they were not kicks at all.
In this context the use of the ball of the foot for most kicks makes perfect sense, and it continues to make sense when used with additional force as impact delivering kicks.
In traditional karate (not kumite competition) performing mawashi-geri
There is no mawashi-geri in traditional karate.
It is a relatively recent development as far as kicking goes, and was introduced along with other kicks like ushiro-geri by the JKA post WWII as karate fighting range increased from around 12-18 inches to 3-5 feet.
If you investigate traditional karate kata you aren't going to find mawashi-geri. You'll find that the more traditional styles use a front kick in it's place and you also won't find it in early editions of karate books which enumerate techniques.
I think when they added it as a technique they retained the requirement for the use of the ball of the foot because of the mechanics of the use of the ball of the foot in front kicks. However. It doesn't entirely make sense in this context due to the risk to the toes, and so the top of the foot or shin is often used instead.
In short, mawashi-geri, along with some others is a sporting karate technique, not a traditional karate one.
Because the ball of our foot is considered the hardest and thickest bone that deals the most damage to opponent's during fight. Take a look at a human skeleton to see for yourself.
