Why? There were a couple of facts I would challenge but they were minor. Otherwise I think what he says is correct.
I don't want to spend much time dealing with this on this thread because I'm writing an article on this topic.
A couple of points though while I'm here.
My understanding is extra calcium in commercial feed had little to do with egg shell quality and more to do with battery hens succumbing to osteoporosis because they prioritized what calcium they had available to produce eggs rather than their own health. If this is true it contradicts the assertion in the video that the hen puts her health before the health of her eggs.
High protein food for some unknown reason has become a thing in western society. There is very little evidence that suggests that we humans in these societies are short of protein eating regular foodstuffs. Should we be short of protein, then the solution is to eat more food. One can get the recommended protein intake per day (varies by demand and age but say one gram per kilo of body-weight for an active person) by eating a couple of portions of steak at say 24 grams of protein per 100 grams of steak, or double the amount of vegetables and grains with a combined protein content of 12 grams.
The same applies to chickens.
The studies I've read and there are a lots of such studies on the net, found that the minimum protein requirement for high production laying hens was around 12 grams per 100 grams. Less than this and egg production dropped off and the hens lost weight with a measured feeding regime. Bear in mind these are high production hens and as such have a higher protein demand than many of the dual purpose breeds favoured by backyard keepers.
16 grams of protein per 100 grams of feed was found to be the optimal percentage, not the minimum. Higher protein content feeds didn't tend to produce more eggs, but did produce heavier eggs and the hens put on weight.
Excess protein is broken down into fats and that is what happened to the hens in the studies; they got fat. This can be better understood by be aware that there is for all creatures a maintenance level of protein required for good health and to make use of protein above this level (to turn into muscle rather than fat) requires exercise usually of a resistive nature. That's why strength athletes require higher levels of protein, the muscle demand it to grow.
A group of mainly confined hens don't get the intensity of exercise required to build past maintenance levels of protein while free rangers often do.
A further problem is commercial feed is digested at a faster rate than wholegrains, some vegetables and higher fiber foods and this makes the hen eat more because she feels hunger earlier than a hen eating unprocessed (natural?) foods.
There is lots more and it gets quite complicated when feeding routines, activity levels, feed composition and availability are taken into account.