Who told you that?!? No, hens lay eggs because that's what they do - with or without a rooster. You only need a rooster if you want the eggs fertilized. Same as in humans - females produce eggs, males are needed to make the next generation. So, boil the rooster and save the hens for their eggs. FYI, at one time I was raising over 100 chickens.
I didn't notice if they boiled or roasted the hen. Boiling would save more of the nutrients and fat. Roasting lets the juices and fats drip into the fire.
Also, they should have had someone pick the meat and skin off the bones and divide it out evenly AFTER the TC. The bones should have been saved to boil again for the broth to make double duty out of it. Did they have rice? Cook the rice in it - yum. Add the feet, as well. Also, break the bones so the marrow can be sucked out. For triple duty, fish with the bones and head. Perhaps fashion a fish hook or fly.
I am saying I didn’t know whether you needed a Rooster for a hen to lay eggs. I suspect they didn’t know either.
Doing a search on the internet shows that a lot of people don’t know and wonder if a rooster is needed for the egg.
Why kill the hen before the rooster...because they thought they needed the rooster for the eggs. It’s their lack of knowledge on the subject - which is not uncommon on this question. That is what I am getting at.