I believe that the tongue isn't doing much in this method, it is just chewing. Our diets in pre-agricultural societies consisted of very fibrous (Cellulose rich) food, a lot of which could be chewed for the nutritional content and then spit back out. Modern fruits and vegetables look nothing like they did in those days (or even 200 years ago), the edible fruit portion has been artificially selected for and the fibrous tissue selected against. An orange did not used to be plump flesh, it was mostly fiber. Same goes with bananas. Here's some examples:

Instead of ingesting all of the fiber every single meal, you could put a large bolus in your mouth and chew out the nutritional content before spitting it back out. This is what Mike is suggesting that you simulate with gum. Make a big wad of gum, and move it around your mouth as you chew it. Hard gum, preferably (Masticah or Falim). Keep your tongue wherever it normally is when you chew (probably moving around the mouth helping to gather up the food being chewed into one part of the mouth). 

What benefit does chewing using "around" the teeth have versus just chewing with the back teeth/molars? You mention mastiha and falim; in this "bolus chewing" Mike Mew says to squash the gum on the roof of your mouth at the end, is this possible with these hard gums or is the consistency just too much (can you "tongue chew" with these gums)?

In my opinion, the benefit of chewing with the whole mouth and not just the molars is to encourage more balanced development. Simple as that.

Now, the whole face is seen as an intricate, interwoven series of muscles. Claimingpower felt this way, and I believe that is a sound principle.
By chewing with the canines, especially the front teeth, and by moving it around.. I believe that muscles will be activated in different enough ways that encourages the WHOLE face. Masseter development is important, yet there are other muscles that can contribute to malocclusion if not kept in consideration.

I argue by chewing with the whole mouth, and applying the pressure along each individual tooth as you move the bolus around that it is advantageous in developing the whole midface along with the masseter. Plus by consciously moving it around, you prevent yourself from going into autopilot chewing which might mean you engage in some form of the habit which caused the malocclusion in the first place.

Tongue chewing with Falim is possible if you use just the 1 stick. Comparatively, you'll need 2 or 3 at minimum to get proper resistance with regular chewing. Falim is a good price, yet the Chios Mastica brand stil gives the best workout imo. However, tongue chewing with anything tougher than falim is quite difficult.

I have integrated this principle into my regular eating by pushing against the sides of the posterior parts of the maxilla & mandible (behind the dental arch) simultaneously as I bite on food. The movement is similar to trying to taste the food with the whole width of the very posterior tongue, especially with the edges near ramus. It got me wondering whether this the functional difference that some of you achieve by open mouth chewing.

Also, chewing with the whole arch helps maintain proper occlusion, so that the mandible keeps up as we expand the maxilla.

It's sort of hard for me to see how our ancestors could have used front teeth to chew in the traditional sense. I think that they more likely used their front teeth and incisors to initially bite into food to break it into smaller pieces. For example, they might take a chunk of meat and rip a piece out using their front teeth. Basically, the job of the front teeth in our ancestors days had the job of utensils such as knives and forks today. It seems to me the front teeth are underused in modern times because the food we eat is nicely cut into small pieces where you can easily insert it right into the back molars; however, without the use of knives to cut pieces of food into small pieces, we need to use our front teeth to break it down initially into pieces that can fit into the back of the mouth where the molars do the rest. Perhaps ripping tissue from the bone also produced a facepulling effect?

Yes ripping meat off bones with the teeth might create some sort of facepulling effect. That might be why, or related to why, certain predators have snouts like dogs/wolves/bears/etc.

As for Mew's 'circle chewing' it seems only useful for gum, because when you chew real food that fills up your whole mouth, you chew with every tooth naturally. I think he actually got the idea from the concept of 'rotary chewing' which is a completely different thing that has more to do with jaw movement and muscle activation than with food or bolus placement in the mouth. Rotary chewing is what cows do when they chew. You know that side to side lower jaw movement?

As for the fibrous fruit comment above, I can't stress the importance of that enough. Chewing tough foods like meats and gums is great for you, but I think that one of the real keys is chewing on crunchy foods. Carrots, celery, etc. All the research I have done into our evolutionary ancestors who had huge huge jaws shows that they were vegetation eaters. And so are most animals today that have jaws much bigger than ours. Think of cows, horses, etc. They don't chew gum or meat but their jaws put ours to shame.

You can just chew up half a stick of celery and leave it in your mouth for a long time, you don't even have to actually swallow it if you don't like to. Crunchy foods gives a great feeling as you chew through them and actually make contact with your teeth on each chew and they let you chew on both sides of your mouth at once which also gives a great feeling. Your tongue also gets naturally activated when chewing real food like this because it has to constantly move the food around in the mouth. I get a good burn in my tongue during these chewing sessions that mastic and falim do not give. Just give it a try.