Yes, the vestal virgins were a State institution along with other offices usually held by government officials.
As for the cause of the fall of the Roman empire, I would say as Tacitus did in the Annals, that the growth of the welfare state and the dependency of the citizenry on that welfare, and the corruption of government officials that goes along with it was as big a cause as any.
Tacitus was commenting on the fall of the Republic. I don’t believe that he even considered the possibility of the fall of the Empire