Birds are efficient animals with a remarkable 65 million years of evolutionary ‘perfection’ that helped them survive the last major/mass extinction on earth. Only one other species also survived - crocodilians.
Part of the birds’ efficiency is lack of a diaphragm which is just one part of what makes them ‘light for flight’. A bird’s lungs are relatively tiny and work more like a bellows rather than balloons. This ‘bellows’ sends air to air sacs located throughout the body. 2 cervical (neck area), 1 intraclavicular (throat area), 2 abdominal, and 2 posterior thoracic (the area at the underside at ‘hip’ location) and 2 anterior thoracic (lower chest/'belly’ area) for a total of 9.
There are many potential causes for an air sac to leak or rupture, including injury from a fall or even a bump at the right angle. Sometimes it’s a spontaneous rupture/leak and no known cause is found. When an air sac ruptures or leaks there’s a bulge at the area which is the air from that sac. If touched it will feel almost like small bubble wrap and make the same sort of “crinkly” sound.
A bird knowledgable vet will, under sterile conditions, simply push a sterile needle into the area and allow the air out. While it may quickly refill repeated releases at least lowers the pressure and discomfort of air being where it shouldn’t be.
Your vet may teach you how to do this yourself - but remember it’s vital that sterile conditions be adhered to every single time.
A course of antibiotics prescribed by your vet is also important ‘just in case’.
In some birds surgery may be needed, but this is the exception rather than the rule.
Sometimes, spontaneous recovery happens, but a vet should at least initiate a puncture to release the air accumulated where it shouldn’t be.
This is especially prudent when the bird is lethargic, going off their food and not behaving the same as the owner knows them to be. When symptoms like this accompany the leaky sac it indicates the bird is likely in some degree of pain or at least moderate to severe discomfort.
I'm not sure I'd return to the vet who did nothing. Was this an actual 'avian vet' ??