Oh, that stinks....and I do...arthritis in both thumbs (thank you to too much flute playing)...and a funky shoulder/back (thank you to horse mishaps and one auto)....so this 50-something body can't reliably hold and jerk at the same time especially not the big cockeral I just put down.
So since I can't swing an axe, slitting the throat is messy and takes an amazingly long time to bleed out especially with sick animals that I don't want to spread stuff with, this works best.
My favorite method for sick, unwanted animals is CO2 created with dry ice, a Home Depot bucket. Put the ice at the bottom, add some warm water, wait for a big cloud to build up (with lid slightly ajar), add bird, wait 30 seconds, it is done and over. Very quick, very painless.
But you can't keep dry ice around, and it is often a pain to have to time a dry ice run when you need to put an animal down.
I can bend and step on a rake still (thank you Lord), so this method works best for me.