My twenty year old cat was doing the same thing as you described to a tee and figured she had mild dementia, she stopped once I got another cat although my other cat was terrible with her, tackling her and trying to mount because he's a young male. I had to put her down about a month ago because her health just kept declining. I would absolutely recommend what you plan which is to put your cat down but within the comfort of your home. One of my biggest regrets aside from getting another cat while my little old lady was still kicking was having to bring her to the vet when that time came. I wish I could have avoided the stress I'm sure she felt, also had to wait a while despite coming early for the appointment which prolonged it and had me crying with her in the waiting area. It sucked.