Then why do they not open it for full public access?
There are lots of libraries that are not open for full public access. I had to get special permission to gain access to the British Library, and, like the British Library, the Vatican Library is
open to scholars and others with a legitimate interest in researching its holdings. The notion that the Vatican Library is closed to everyone but the pope and his minions is a fabrication perpetrated by anti-Catholic conspiracy nuts and third-rate writers like Dan Brown.
It's not secret in the modern sense of the word, but it's practically impossible for anyone to go in and see what's in there, and any researchers who want to see something that's in there have to ask for specific documents.
That's pretty typical of any kind of archive. You can't just walk in and rifle through the stacks.
Due to the long and rigorous application process few requests are granted entrance.
As well it should be. Otherwise every crank in christendom would want to get in.
For all practical purposes this means that the secret archives remain closed off and it's secrets unexplored by any except it's keepers.
Except for about a thousand researchers each year who are allowed access
It is fair to assume that had they just said "no, no one's allowed in" the general public would not stand for it. So instead they have just made the process of gaining access so backward and difficult that the end result is much the same.
The general public has no say in the matter. And for catholics, that's par for the course, so they're probably not complaining.