Unfortunately I have a bunch of experience with things popping out of socket on puppies since my 3 month old bulldog suffered an undiagnosed dislocated hip socket because of severe hip dysplasia. That was 2 months ago and things are going good now but here's a few things I can tell you after seeing two orthopedic specialists, one bulldog specialist, and my regular vet (and receiving a BUNCH of support and advice on this board, which cannot be understated):
1. X rays in a puppy that young are very difficult to interpret. No one on the internet will be able to interpret them for you and , in fact, most vets cannot interpret them. Puppy joints are loose and seldom resemble what they will look like in later life. They should be looked at by a radiologist before any surgery, because they have gone to school 6 additional years to study the science of reading x rays and they may be able to make a more accurate diagnosis, especially if you're considering a drastic procedure!
2. You need to see an orthopedic surgeon. Take your x rays with you so you don't have to get more but they need to manipulate the joint, see the dog walk, and perhaps even take more x rays at different angles to get a better look. They have gone to school 4-6 more years that a regular vet and possess expertise in these things that cannot be obtained at a vet's office. I am shocked any orthopedic specialist would offer a diagnosis based solely on the x rays, that is incredibly irresponsible so I would find a different one, preferably one that has some experience in the highly unique bulldog build.
3. Arthritis is an issue in any joint that is not perfectly formed however it does take some time to develop. Our bulldog went from 10 weeks-14 weeks with a hip partially out of socket - so a month - and had a perfectly clean socket so it takes time, as in years, to develop. So although a "wait and see" situation is frustrating, in the long term, it can help the vet make a better diagnosis by allowing more time for the puppy to grow and bones to form.
4. Do not do anything irreversible, except as a last resort. This is really important and I am absolutely shocked an amputation was even on the table right now. In my puppy's case, the first thing the surgeon recommended was a little time, then, when she took more x rays and recommended a procedure to remove the dog's femoral head, she called in the middle of the procedure and said she thought she was able to anchor the hip back into socket and give it time to grow in - as opposed to having the femoral head removed. This is the option we elected to have done and although there is a risk the hip (or in your case, elbow) might pop out, there is probably a good chance it can be anchored back in through surgery.
I hope this is helpful but please, go visit an orthopedic surgeon. I am certain there are many alternatives to amputation if the initial diagnosis is correct.