I know of no evidence that finding a cure to any certain disease would mean immortality. If there was such evidence then I'd be in favor of pursuing that research. Nothing is worse than death except being alive without being able to take meaningful actions in relation to the world around us (i.E. I would not want to be immortal if the price was that I would spend an eternity with dementia).
Yes, we should fight specific diseases with the objective of figuring out immortality, because most people want to live longer. There is nothing wrong with wanting to extend life. Fighting disease is important, because a good number of people die from specific diseases, rather than old age. Specific treatments could cure cancer or heart disease.
The goal of medical research into the causes of and cures for specific diseases should not be finding a path to immortality. The goals of this research should instead be finding economic treatments and cures for these diseases. Society should have medical research goals that are of benefit to all of mankind.
I think the idea that a human being can be immortal is ridiculous in the first place. And we definitely should not put more of an emphasis on one disease over another just because we think it will help us a species to figure out how to find the answer to immortality.
I don't believe humankind will ever achieve immortality, or even that it would be an ideal situation. Instead, disease treatments should be done in order to prolong life and increase comfort for those who are suffering. This will likely never lead to immortality because the body's cells are really not capable of that. But having the goal of alleviating pain as much as possible is still a good one.
We should not fight specific diseases with the objective of figuring out immortality. Specific diseases should be fought to preserve the quality of life for people who suffer from painful diseases,and also to increase the life-span that would be shortened if the disease was untreatable. The objective of fighting disease should be to promote a healthy existence, not to figure out immortality.