Mind you, this is theory because I haven't put it into use yet. But have started building a set-up similar to the one in this
link. Not as pricey, pretty, or shiny - but essentially a budget model. It doesn't need to be chrome with spinners to do the job afterall.
As far as reducing stress on the birds;
First I plan on culling in batches in the evening. The quail obviously don't go to roost, but they tend to settle down quite a bit at least after dusk. Each batch I do, I'll move them into a holding cage the day prior. Building my CAK chamber to be able to hold said cage so that I can just slide the cage into the chamber.
I'll pre-load the chamber with the gas. Some will obviously escape when loading the cage, but it should still house enough that a refill should be quick. I imagine the quail to go quickly from size estimate alone. For chickens, I guess it's simple math as I won't be loading as many chickens in the chamber at a time. I expect to cull fewer than 2 dozen chickens this summer, but imagine the quail could reach many dozen.
Realistically, I don't know that the CAK system is better/easier/more efficient than wringing but I also don't know that it is less so (for the birds). It may be easier in that it's less individual work for me, and potentially less mentally taxing in that you're not extinguishing each individual bird in your hands. I hope that's the case, but obviously won't use a flawed machine if its apparent that it is prolonging the procedure. What I can do, is post about my findings here later this summer and share the designs for the system if it appears to be practical.
I came across this a while back, and was sort of a loose inspiration for the CAK system. Viewer warning, the video shows the euthanasia of a pig via inert gas, do not watch if you don't want to see it..