Which museum will be the next to appoint a female director? Most of the world’s great visual arts institutions remain firmly under male control. The current bosses of the British Museum and the National Gallery in London are men, as are the directors of the Prado in Madrid, the Louvre in Paris and the Metropolitan Museum of Art in New York. So which groundbreaking top-rank art museum will be run by a woman from 1 January 2017?

It is one of the oldest and grandest of them all, and you will find it at the heart of perhaps the most male-dominated enclave in Europe. The new director of the Vatican Museums, whose collections started to be amassed in the Renaissance, is the Italian art historian Barbara Jatta, who previously curated prints there and rose to be deputy director. She has now been appointed by Pope Francis to succeed the current 77-year-old director, Antonio Paolucci, in the new year.

This is not as surprising as it seems. The Vatican may be one of Europe’s last surviving medieval city states, run by cardinals and policed by Swiss guards with halberds (at least that’s how tourists are likely to see them – they have guns and tanks too), but its incredible complex of museums has a history of unexpected modernity. What other religious institution collects atheist art? The modern gallery of the Vatican Museums proudly displays a grotesque painting of a suffering pope by the defiantly godless painter Francis Bacon. That’s probably the most provocative presence in a sensitive and imaginative collection of modern religious art that also includes Van Gogh’s Pietà (after Delacroix) and paintings by Dalí and Picasso.

The Last Judgment by Michelangelo, inside the Sistine chapel. Photograph: Alamy

The Vatican’s museum is so vast and diverse that it is a plural. It is called the Vatican Museums and consists of a connected sprawl of collections that fill a large part of the Vatican itself. Queues often stretch right along the outside of the papal palace complex. Once you get inside, the cornucopia of art includes the Pinacoteca, where such paintings as Leonardo da Vinci’s St Jerome and Caravaggio’s The Entombment of Christ hang; numerous archaeological collections that have everything from giant bowls from Nero’s palace to an extraordinary Egyptian art installation from Hadrian’s villa; and a fine museum of global art.

A recreation of the Raphael Loggias at the Hermitage Museum in St Petersburg. Photograph: Dave Bartruff/Getty Images

At the heart of it all is the Belvedere courtyard, one of the oldest extant art galleries on earth. Here stand some of the most renowned masterpieces of classical sculpture including the Apollo Belvedere and the Laocoon. For centuries, this was the most hallowed destination of the Grand Tour. Young aristocrats from all over Europe came to see the Apollo Belvedere, considered in the 18th century to be the most beautiful statue ever made. After paying antiquity its due reverence they made their way, as the crowds still do today, towards the climax of the Vatican Museums – through the rooms of Raphael to emerge into the glory of the Sistine chapel.

It is great that such a majestic and venerable museum has Jatta as its new director. But what should she do to modernise it further? Can such a marvellous collection be improved? In recent years, the big issue for this museum has been to deal with its colossal crowds in spaces that were originally designed for cardinals and popes to spaciously hang out in. Measures have had to be taken to preserve the Sistine frescoes, at risk from the sheer number of people passing through. Jatta will no doubt have to wrestle with these problems just as Paolucci has.

Yet I have a suggestion that, as well as enriching this great complex of museums, might also alleviate overcrowding, simply by providing more spaces to explore. It seems incredible that with so much art and architecture on display the Vatican still has hidden treasures – but it does. Hidden behind closed doors tantalisingly near the museum are Michelangelo’s paintings in the Capella Paolina and Raphael’s Loggia, his greatest work, whose opening might help us understand much better why he is one of the greatest of all artists. In the 21st century it seems archaically exclusive to keep these glories for the eyes of the Vatican elite only. The same progressive spirit that has given the Vatican Museums a female director ahead of comparable art collections must now democratise public access and open up this clerical city’s hidden spaces. That would be a great cultural demonstration of the Vatican’s readiness to change.