But I keep feeling guilty, like I could take him to the vet instead.
I know some people treat their chickens as pets, and a vet visit might be something to consider. Where I live, the vets don't know much about chickens, and it would cost you $$$ for nothing. I would never consider taking any of my chickens to a vet. For the price of one vet visit, I could replace my entire backyard flock 3X over! Not kidding.I would much rather harvest an injured chicken myself. If a bird is sick, however, I will not butcher and eat it.
I don’t know what’s wrong with me.
Having the ability and skill to process your own chickens is a great advantage. Many people have the opposite problem of yours and they cannot get themselves to butcher their chickens.
Obviously, I don't know all the small details between you and your mother, but it sounds like you did the right thing to me. Assuming, of course, that the rooster was yours and not your mothers....

Have you considered getting meat chickens that must be harvested around that 10 week old period? Knowing that you are raising chickens for an early harvest may help you get over your bad feelings about butchering a chicken. I have lots of respect for the chicken's life I take for their meat, but I don't feel bad about it. I think because I started with meat chickens and had to butcher them myself at 10 weeks, it has prepared me better to butcher a laying hen or rooster if/when the time comes. Having said that, I know it is more difficult to take the life of a bird that you may have cared for many years.