Anyone with a simple knowledge of the history of western civilization understands the special status of the Vatican and why it is treated as a state. It has more historical legitimacy than many other nation-states out there.
Its independence was granted by Mussolini, so no in my view it is not legitimate unless you consider Mussolini legitimate.
If Italy re-granted the Vatican's "independence," would that make it all better? In many ways, the Vatican is an ersatz country anyway. It is a complex with nice gardens, with a few permanent residents. Most folks "commute" into it for work. It is about as independent as Monaco in some senses. It exists based on the tolerance of its host. And finally, almost nobody cares about this issue for good reason. It just doesn't matter.