One of the arguments in favour of The British Museum refusing repatriation requests is that it is not the British Museum, but in fact the World Museum. I would say that if there were to be a World Museum, from a European perspective, I would expect/want it to be somewhere like London, Paris, or Rome. Or Beijing and New York outside of Europe.
Well it just so happens that the biggest/most visited museums in the world are The Louvre (Paris), The Nat. Museum of China (Beijing), The Vatican Museums (The Vatican), The Met (New York), and The British Museum (London), each with over 6,000,000-9,000,000 visitors per annum. Compared to museums in Rome proper, such as the Castel San Angelo, which only receive 1,000,000 visitors per annum.
What's going on here? Are The Vatican Museums seen as Italy's national museum, despite not being part of Italy? Why does Italy not have a prominent national museum, despite the nation's rich history? There's The National Roman Museum, but that is only concerned with a small fragment of the nation's history. Britain doesn't have the national Roman Museum, Tudor Museum, Norman Museum, Colonial Museum etc at the expense of the national British Museum. So how did this happen? I'm also fairly ignorant of the history and of the relationship between Rome and The Vatican, so any broadening of that understanding by someone knowledgeable would be really nice.
There most certainly is a National Museum in Italy, however it is designed and managed as a system of interconnected museums rather than an a single unifying structure. While the National Roman Museum (Museo Nazionale Romano) is one of Italy's most important museums, spread across four disparate exhibition spaces (the Diocletian Baths, housing prehistoric art, general inscriptions, and modern art; the Palazzo Massimo, housing ancient art, jewelry, and coinage; Palazzo Attempi, focused on renaissance-era archeology, and the Balbi Crypt focused on Medieval Archeology) there is a fundamental thing to keep in mind in that most artifacts in Italy have remained largely where they are, or have otherwise not moved far from their original places. This is true from artifacts from the Roman Era, many of which have been already collected and exhibited since at least the Renaissance, to art from the Renaissance itself (and beyond), which is largely still located in or near its original location (it would be difficult and unpopular, to say the least, to move Michelangelo's David from Florence to Rome). Thus collecting artifacts of national interest into a single chronological exhibition documenting the nation’s history is not necessarily possible or even desirable, since not all artifacts can be moved in the first place: the Ministry for Cultural Goods and Activities (Ministero Per i Beni e Attività Culturali, which I will call “Ministry of Culture” for brevity) does not have blanket permission to sieze and move artwork and artifacts which might be located in museums managed by combinations of regional, provincial, or public-private authorities, nor would it be particularly productive to try to do so, as it would depauper museums across the country. Indeed, an opposite approach is taken, whereby even in those museums which fall within the “Musei Nazionali Italiani” system, that is to say the hierarchy of museums managed (in whole or in part) by the Ministry of Culture, the philosophy is to exhibit the cultural heritage of each individual region through the national museum system.
As an example, while the largest National Museum on pre-Roman Southern Italy is in located in the city of Taranto, an extensive National Museum network also exists based in in Reggio Calabria, with satellite locations in Crotone, Locri, and other parts of Calabria, each exhibiting local archaeological finds, as well as acting as conduits through which ongoing excavations, educational initiatives, and university-affiliated programs are managed (just as is done in the Museum in Taranto). Each museum leverages the geography, history, and the ensuing wealth of archeological work taking place specifically in their region. However, important albeit smaller exhibitions on pre-Roman archeology are also displayed in, just as an example, the two archeological museums in Abruzzo located in Campili and Chieti, where a plethora of art and artifacts linked to the Abruzzo region’s pre-Roman Italic Tribes are located (and these two museums fall under the management of the Ministry of Culture’s directory for in the Abruzzo region, overseeing fifteen other institutions classified as National Museums in the region, each focusing on a different aspect of the Abruzzo’s history). The museums in Chieti and Campili are both denominated “National Archeological Museum of Abruzzo,” and are in every way a “National Museum.” Thus the same philosophies are reflected in Abruzzo just as they are in the National Museums in Reggio Calabria and Taranto.
Some museums do display collections of items spanning a wide stretch of epochs: the Museum Circuit of Piazza San Marco in Venice houses in a single museum structure both a National Archeological Museum exhibiting an eclectic assortment of ancient greek and roman art, sculpture, and coinage acquired through both local archeology and donations from local collections (the first of which was bestowed in 1596!) as well as the Correr Museum of Venice, which is overseen by the Foundation for Civic Museums in Venice (a publicly-owned foundation managed by the City Council in Venice, and thus not the Ministry of Culture in Rome or one of its regional directories) and houses art and artifacts dating from the renaissance through to the 19th century, and even hosts temporary exhibitions of modern and contemporary art, thus offering a truly thorough circuit documenting the material history of Venice and its surroundings. A similar example are the Museums of the Sforza Castle of Milan, which present a circuit running from the Ancient, Medieval, and Renaissance art from Milan, in addition to smaller exhibitions constructed around individual donations ranging from Ancient Egyptian artifacts to modern furniture design.
However, this sort of all-encompassing circuit is often an exception rather than the rule, and is seldom present in museums under the National Museum system, since as stated above, the objective of National Museums is to act as a network, rather than single collection point. The Museums of the Sforza Castle of Milan mentioned above, for example, are managed by the Directory of Culture of the City of Milan, analogous to their Venetian counterparts managing the Correr Museum, and do not fall under the National Museum system.
The methods by which the innumerable museums outside the National Museum system came to be are varied, and when they are not focused in art and artifacts from Italy, their creation is almost always circumstantial: In Milan, the 19th century high bourgeoisie developed an interest in anthropology and ancient history (with a particular interest in egyptology) much like the bourgeois in many other parts of Europe, but given the smaller size of the city of Milan compared to the other bourgeois capital of Europe, as well as the Italian Nation’s lackluster performance in the colonization process compared to their European peers, meant that collections eventually donated to museum institutions in the country were small. In any case, artifacts donated to the Milanese museum system from cultures outside Italy are displayed in a contemporary structure dubbed MUDEC, and thus are managed by the City of Milan with little organizational oversight from the Ministry of Culture (on the specifics of Egyptology, the largest exhibitions on display in Italy are actually in Turin and Florence, where pre-unification monarchs developed a concentrated interest in collecting items and artifacts from Ancient Egypt, but nonetheless did not rival the voracity by which American and British collectors extracted artifacts from Egypt).
Private associations and foundations, often working in concert with local authorities, can also contribute to developing museum structures: an example is the Museum of the City of Bologna, founded in 2012 by a private foundation. This is an elaborate museum structure narrating the history of Bologna through the Etruscan, Roman, Medieval, and Modern eras, collecting, acquiring, and restoring artifacts and artwork from the city’s history. Certainly, as you mention, among these non-governmental organizations the the Catholic Church and its affiliate organizations is a large collector and exhibitor of historic art and artifacts, and is by no means limited to the (admittedly very large) Vatican Museum in Rome: the Ambrosian Library in Milan, just to name an example, manages an art collection that rivals both the Sforza Castle and the Brera Art Gallery (the two largest museums in the city).
In the spirit of the decentralized National Museum system, there is no impetus to unite these various museums, be they private, or managed by regional, provincial, and city authorities, into a single exhibition space.
Indeed, given Italy’s history as a divided nation, the conditions to construct a museum collection which might approximate a national narrative only emerged after the country was unified in 1861, by which point several museums already existed (mostly in and around the courts of the recently-annexed monarchies on the peninsula): An archeological museum existed in Rome flanked by two other museums (dedicated to Prehistory and Etruscan art); as well as similar archeological museums predating unity in Naples, Florence, Cagliari, Taranto, and Parma; and not to mention other museum structures housing non-archeological collections of art and architecture under a multitude of ownership formulas. All the aforementioned museums are still open and operating as National Museums today, and indeed after unity the philosophy was never to construct an all-encompassing exhibition space, but rather develop a model of local museum structures all across the peninsula: Post-unity archeological museums were opened in Palermo (1866), Syracuse (1878), Ancona (1908) and Matera (1911). Following the Second World War, a broader spectrum of museums was brought under the Ministry of Culture’s umbrella (and new museums were constructed), creating the complex network which is in place today.