Hi Colin,
This is most likely an age related issue. It could be related to nutrition or internal organs that are not fully functioning anymore. I had a very old lovebird that grew a tail feather like this the last year or two of his life. Instead of the old feather molting, the new feather would start to grow in, fused to the end of the old feather. This resulted in a chain of feathers that would eventually fall out. There are certain Avian diseases that birds can carry without symptoms, that can cause problems when they get elderly. You can try a better food or a trip to the Vet, but with an elderly bird, big changes are not always the best. If this is a normal part of his aging, it would be a shame to stress him out for a trip to the Vet or try to force him to eat better. If he is happy and eating well, I would be inclined to let him be and enjoy his final years, however many that may be. A parakeet can live up to about 20, but most captive bred budgies only live about 6-8 years due to poor genetics or poor nutrition.
Thank you for asking Lafeber,
Brenda