Why are holy religious cities not their own city states, the way the Vatican is for Catholics?

Why is the Vatican the only one that is a city state?

Because what we know today as Vatican City used to be the much larger Papal States, which ruled much of Central Italy... long story short, by 1860 it had lost a great deal of its lands to the upheavals in favour of an unitary Italian nation - and its fate was sealed by the defeat at the hands of the Piedmontese during the battle of Castelfidardo.

The territory of the Papal States was then limited to Rome, as well as part of what is today's Latium; its independence was guaranteed by France and it was exactly in order to avoid a war with that country that Garibaldi's 1867 expedition was stopped short at Mentana! But what was left of the P.S. was now a shell of its former self, and the Italian government had begun expressing its interest in moving the capital to Rome more than once - so, in order to put pressure on both the Pope and the international community, it saw fit to relocate the capital to Florence in 1865. Five years later - with France too busy fighting in the Franco-Prussian War - the Italians took the opportunity to finally capture Rome (20 September, 1870) and proclaim it as the new capital of the Kingdom... however, the Pope did not recognise the new state of affairs and declared himself a "prisoner in the Vatican", forbidding Italian Catholics from participating in the new country's political life.

Despite the Italian Parliament having already passed the so-called Legge delle Guarentigie (which affirmed the sanctity and inviolability of the Pontiff's person, granted him the right to receive foreign ambassadors and established a public fund for his expenses) relations among the two parties remained strained until the early 1900s, with the election of the more liberal Pope Benedict XV; such tensions were evident in the markedly anticlerical stance the Italian governments had adopted up to that time, as well as the urban fabric of the "new" Rome - whose first Town Planning Scheme (piano regolatore) of 1873 was a veritable monument to such aspirations. In short:

Among the new districts built after 1870 (Savoia, the Esquiline, etc...) Prati had its road layout designed in such a way that a pedestrian could not see St. Peter's dome from its streets despite it being right next to it.
Many religious orders were suppressed and their properties torn down, sold for redevelopment or otherwise used for new purposes (e.g., Rome's oldest jail - the Regina Coeli - was originally a Carmelitan convent).
Roads, statues and other monuments were dedicated to famous people the Church had been in contrast with throughout history - a lungotevere was dedicated to the proto-patriot Arnaldo da Brescia; a monument to Giordano Bruno unveiled in piazza Campo de' Fiori; the main thoroughfare leading to the Vatican (via Cola di Rienzo) dedicated to a 14th-century Roman who had revolted against Papal authority; the Janiculum hill dotted with busts of all those people who had died defending the Second Roman Republic of 1849; a new monument to Garibaldi (whose horse's buttocks faced St. Peter's!) placed on top of that hill; a grandiose ossuary containing the bones of the patriots who had died in 1849 placed nearby, and so on and on and on...

Anyway - the whole matter, known as the "Roman Question", was a source of nuisance and slight embarrasment for the Italian government: it would only be settled on 11 February 1929 when Benito Mussolini, in a cunning political move aimed at pleasing the more religious strata of the population, signed the Lateran Treaty which established the modern, independent Vatican city-state as we know it today.