I'm posting with the hope that people can give input and insight on if/how I should push my vet towards a hind leg amputation for my dog.

Backstory:
My dog is a 10 year old Doberman, and completely healthy except for his back left leg. He didn't show any signs of discomfort until three years ago. He began having issues jumping up onto the couch, would get sore after playing fetch, etc. So, naturally, I took him to the vet to get checked out. They did X-Rays, and incorrectly diagnosed him with bone cancer. We were referred to a specialist, who correctly diagnosed him with degenerative arthritis in just his back left hip. The reason for the arthritis was determined to be the result of a shortened femur. The vet guessed that he had broken his leg before I adopted him when he was about a year and a half, and that the broken leg hadn't been properly set. It grew back slightlly bent and short, which over time put more stress on his hip, and has led to his current arthritis issues.

Current situation:
Since getting the correct diagnosis, he has been on daily pain medication. The arthritis is degenerative, so it will only get worse. We've gotten to a point where he clearly isn't comfortable even with the pain medication. Last year, I brought up the idea of amputation with the vet and an orthopedic specialist. Neither was particularly supportive. The vet wants to keep giving him pain pills. The orthepedic surgeon wants to do a full hip replacement and femur extension to the tune of $10,000 and no guarantee of it actually working.

The hip replacement is completely off the table for me for a variety of reasons including cost, recoverY time, my dog's age, and the overall probability it will help. I'm continuing to give him pain medication, but I can't help but thinking he would be way better off if the leg was just amputated. In my mind, the only thing that causes him pain is the leg. It would be a win-win for him to have it amputed. He gets to be pain free and wouldn't have to take pain pills. I'm not sure why both vets didn't advocate for this. Neither outright objected to my position...they just were pushing in different directions. I want to do what is best for my dog, so I don't want to push for something that is a bad idea. I understand they are the experts, but am I crazy for thinking an amputation is a good idea?

Does anyone have any insight into this type of situation?