Well, for one, we know there are many ancient treasures in the Vatican even if we don’t know exactly what they are. And for two, the Vatican has actually be extant throughout the period since the fall of the Roman Empire, while the national government of Italy has not.
In my culture, we call this “unfounded conjecture”.
Not quite -- while the Church administration has been extent throughout the period, the site of the current Vatican has not
The Popes lived at the nearby Lateran Palace (which is in Rome proper - on the opposite side of the Tiber) until 1350 and their return from Avignon, when they moved to the current Vatican site.
The current St. Peter's basilica was built on top of the old St. Peter's basilica (built in 309 ad) which was built on top of St. Peter's grave. It was not a place where "treasures" were kept
Also we KNOW that the Menorah was sent to Constantinople at the time of Justinian. There is little to no way it was brought to the "lost west" after that.