I had to struggle with this issue with 2 of my birds.
One pigeon lost the use of one leg.. it is a king pigeon.. so a heavy breed. It spend most of the time sitting down and doing nothing. I felt very bad for it.
The second bird is a bantam rooster. He got either some spinal injury or disease that affects the nerves. He could not stand, walk and he had to use his wings to balance. He lost weight and did not have much strength to lift up his tail or head for long.
Now the good news.
The pigeon has regained partial use of the leg.. it has built up muscle in its other leg.. It is now back to normal activity level.. hopping around easily and when perching it can use the bad leg a little and use the foot on that leg to grip the perch and roost fine. It seems very happy, bathing and acting normally.
The rooster very gradually improved. He can now walk, run, and stand up without collapsing. He no longer has to use his wings to support him. He is crowing and mating with the hens like normal and has put on body weight again. However, he now stands on his tip toes.. and when walking will kick each leg right out in a high step motion... But he is very happy, in not pain, and can do all the normal chicken things with ease.
My point is give any injured or sick 'pet' birds some time to heal or adjust. Like us humans if we have an accident or become disabled in some way.. at first its hard to adjust.. but over time we can adjust and cope with life.
If the birds is in pain, getting worse, not getting more mobility, or is being picked on by other birds, then its best to cull them. Just don't be too quick to give up on them!