I am a veterinarian and while declawing is not as popular as it used to be, people still request it. And yes, it involves amputating the tip of each toe at the knuckle so that the growth bed for the nail is removed and the nail does not just keep growing back. I know it is illegal in the UK and Australia, as well as many other countries. Here in the US, people seem to accept it as necessary when their cats become more destructive than they can live with. In some cases, owners cannot afford to get scratched and have it possibly get infected (they are diabetic, or have a compromised immune system in some way) so declawing the cat is an option. Unfortunately, some people are semi-forced into having it done because they have a lease agreement that requires that all pet cats be declawed.
The surgery is done under general anesthesia. When I have performed it, I have used every method of pain control at my disposal (including locals, opioids and NSAIDs - some vets are not as proactive when it comes to pain managment). Most cats seem to do okay with it as long as it is done at an early age (often at the same time as spay and neuter). I prefer that it be done as a last resort - when trying to train the cat to use a scratching post, keeping the claws trimmed, etc. is not working. I would agree though that it is not natural and try to get owners to think long and hard about their reasons before deciding whether or not to have it done. I work in emergency medicine now so I am not expected to do declaws as an elective surgical procedure. Some vets will argue that it is better to declaw a cat and keep it indoors than to give it up, or kick it outside.
Interestingly, I am originally from Michigan, moved out to New England after finishing vet school, practiced out there for 11 years before coming back to Michigan. People do not declaw their cats out there nearly as much as they do here. Out there, I might have done 3 or 4 a year. In Michigan, when I was in general practice, it was 3 or 4 a month. Out there, people would ask me about it but feel terribly guilty about having it done (and yet also feeling like they had no choice). In MI, many people don't seem to give it a second thought. They adopt a new kitten, it's 8 or 9 weeks old and they want to know how soon they can have it declawed. And they want all 4 paws done (which I won't do) instead of just the front paws which do 99% of the damage. When I ask them why, they just respond that their other cats have always been done this way. It's like they can't even consider the alternative. "Because our other cats have always been" is not a good enough reason for me. There have been some people that wanted their cats declawed (like 1 girl who wanted her 12-year-old cat declawed because they were getting new leather furniture and did not want it ruined) where I have flat-out refused to do it because I did not believe it was in the pet's best interest - it was strictly for the owner's convenience. It surprises me that there can be such a different attitude from one part of the country to the next. I believe one community in California tried to ban declawing a few years ago but it didn't stick.
I do wonder how cat owners handle their cat's clawing behavior in countries where declawing is not an option (and in many of these same countries, cropping of ears and docking of tails in dogs is illegal as well - something that is still done routinely in the US strictly for appearance).