Chicken bones have a tendency to 'splinter' when 'mashed' (as in chewed upon), which is why you never give chicken bones to a dog.
This applies equally to humans, if we gnaw on a chicken bone it is more like to create a harmful splinter that may find itself lodged in any number of places in your digestive system.
That said, as has been commented above, extracting the flavorful marrow and those 'last little bits' from the carcass can be achieved by making stock.
If you are wanting to reduce the waste, I would then recommend drying and grinding the remaining bones to provide calcium into your compost.
[Edit]
Interesting comments from @JohnEye
& @Molot prompted a bit of research.
Here is an article from WebMD that supports my original statement.
Is Eating Chicken Bones Bad For Our Health
Other articles suggest that some people do this, but I would still conclude that it is not safe to do so, though apparently it can be done.
cooking.stackexchange.com/users/6279/Cos Callis
Pressure cookers will quickly soften most chicken bones.
We make stock with our chicken carcasses in a pressure cooker, and the resulting bones can be crushed with fingers, no splintering.
cooking.stackexchange.com/users/1129/mskfisher
Absolutely!
My wife makes her own dog food out of 1 Lb of rice, 1 Lb of carrots, and the trimmings of 1 whole chicken.
After she breaks down the chicken for the meat that the family will eat during the week, all the trimmings, including the bones, are put into a pot and boiled so we get the marrow and gelatin from the bones as well.
I'd describe it as making stock except that the liquid is not separated for reasons I'll get back to.
Once the 'chicken stew' has cooked, she finds all the bones and puts them into a high-end blender to render into a thick paste we call a 'bone shake'.
She puts the rendered bone into the the stew / stock along with the rice and carrots to finish cooking.
Once that is done
she adds vitamins and other supplements to the batch before dishing out portions for freezing
and then we have dog food for several weeks.
We have used everything from the chicken, including the skeleton .
We joke that the dog eats better than we do and have had multiple vets give their approval.
Now, for those who ask what this has to do with human food, my response is even though we use this process to feed the dog, there is nothing special about it and would be perfectly safe for humans.
The bones are completely rendered in the sense of cook time and safety.
Cooking them in the stock makes them safe from a bacteria perspective and using the blender essentially grinds them into particles so small as to eliminate any issues with splintering.
In fact, my wife started to make (dog) cookies by adding sugar, flour, salt, etc.
I mention this to illustrate the multitude of uses for the bones so there is no reason why the bone shake couldn't be incorporated into other recipes for people.
cooking.stackexchange.com/users/1783/Kelly S. French
We recently made a chicken bone stock by cooking a carcass in water with a few tablespoons of vinegar.
When it was done, the bones had demineralized enough that they could be chewed thoroughly without any difficulty nor observable splintering.
cooking.stackexchange.com/users/67638/a.o.d
Torigara (鶏がらスープ) is a common Japanese recipe most commonly associated with the making of ramen.
The recipe usually calls for a whole chicken carcass or chicken bones and using a very long cooking time (eg.
5 to 10 hours)
in order to reduce everything to a rich broth that can be made into a ramen.
Some recipes call for discarding the bone while I am aware of some ramen shops actually pounding the soup and keeping the crushed bones (which renders down to powder) as part of the soup.
The recipe is common in ramen shops all around the world and is safe to consume.
PS, I think the addition of rice-vinegar in some recipes would help breaking down the calcium in the bones faster.
cooking.stackexchange.com/users/67664/user67664
I cooked my whole turkey carcass for several days and poured all the bones and juice from boiling it, a little bit at a time: liquified it in my osterizer to get all the marrow and nutritional value but strained it to get rid of all the pulp at the bottom and have been drinking it.
It’s delicious!
I’ve heard it’s good for arthritis.
It made over a quart to drink!
cooking.stackexchange.com/users/80023/Lyn Leonard
Using a pressure cooker and a vitamix blender you can turn the leftover bones into a paste.
cooking.stackexchange.com/users/74081/Tom Balint
I like to preserve as much as possible from my cooking and trash as little as possible.  With that in mind:   can chicken skeleton be made edible?  or is there a way for the bone marrow to be extracted and cooked?   Or does it exist a reason for why it is a bad idea to consume chicken skeleton?
Can chicken skeleton be made edible?
cooking.stackexchange.com
<chicken><bones><marrow>
cooking.stackexchange.com/questions/90338
cooking.stackexchange.com/users/67609/Svintoo
The easiest way is to make stock by cooking the carcass in water with a few tablespoons of vinegar. Chicken and turkey bones can be cooked in pressure cooker, which will soften them enough to make a paste that can be used or strained and drunk. Chicken trimmings can also be mixed with rice and carrots to make dog food. It is generally considered possible to make chicken bones edible. One method id to make a stock. Moreover a pressure cooker can be used to soften the bones quickly. Poultry bones can be liquefied in an osterizer. They can also be made into dog food.
Chicken bones can be made edible by making them into stock. Pressure cookers can cook the bones until they are soft enough to eat. Dog food can be made from chicken bones. Poultry bones can be boiled and liquefied in an osterizer. Yes chicken bones can be made edible.