I tried to do this myself - I didn't want to burden my husband, I just couldn't bring the axe down. I didn't feel confident I could do the method you mentioned either.
Like I said, I learned how to kill and butcher some animals before I ever learned how to care for them. In that respect, grandpa used to go duck and goose hunting. When a bird was shot, they were often injured but not yet dead. Out in the field, you wrung the neck of the bird to kill it and prevent further suffering.
I had no problem wringing the neck of my sick chicken to end its suffering immediately. It's fast and effective for both you and the bird.
We consider them pet and egg providers - and are emotionally invested, even though we know the birds are fragile and expendable - and are really here to provide.
It's more difficult to put down a pet. I can understand that. Even though my hens are not pets, a person does get emotionally invested in an animal you have taken care of for years. I try to keep an emotional distance from my hens knowing that they don't live long. But they are well taken care of and have a good life until the end.
We would not get meat chickens unless we had a butcher to manage that portion - I know my limits.
Yeah, I got meat chickens first because I wanted to know if I could butcher them after 10 weeks. Frankly, it was more work than I thought it would be. I only had 50 meat chickens, but it was a lot of work to butcher and pluck the birds without a machine plucker. We only had meat chickens for 1 year. Since then, it's been just laying hens.
In theory, I would want to butcher my hens after a couple of years when their egg production drops down to a level so low that it does not make economic sense to carry them any longer. But, so far, I have had a few chickens die from some sickness. I won't butcher a sick chicken. They just get tossed out in the tall grass and returned to nature.
You might want to consider getting some meat chickens that you will butcher at 10 weeks. I think you find that if you raise some chickens with the notion that they will be food in 10 weeks, you would not treat them as pets, you will have an emotional distance from them from the start, and the act of butchering them would be easier to perform.
Once you have the butchering skills for meat chickens, you could use that same skill set on butchering laying hens at the end of their life. You might be more attached to the laying hens, but at least you could fall back on your skill set to butcher the chickens and maybe put your emotions in check for a while.
It sounds like you know your current limits, but you can expand them. You don't have to accept being stuck where you are. Especially if your goal is to have chickens to provide both eggs now and meat in the future. I wish you the best.