there are two major facts that led to pizza becoming THE dish to be ordered via telephone and delivered to the home.

After World War II, the telephone networks saw rapid expansion and more efficient telephone sets, such as the model 500 telephone in the United States, were developed that permitted larger local networks centered around central offices. A breakthrough new technology was the introduction of Touch-Tone signaling using push-button telephones by American Telephone & Telegraph Company (AT&T) in 1963.

While, at the same time an estimated 600,000 of Italians arrived in America in the decades following the war. As was, and still is a regular occurrence for immigrants, a large number of them ended up working in the food industry, often opening inexpensive restaurants serving Italian food.

Which was a fortuitous choice, because not long before, the American veterans of campaigns in Italy returned home, with a taste for Italian food already developed.

As it were, first inexpensive and proliferated telephones, and the influx of cheap, tasty Italian food was a match made in heaven. The startup family restaurants ran by Italian immigrants could not compete with established food industry, could not afford rent on big floors required for a major restaurant to sit enough clients to support themselves, But they could deliver the food to the homes of their patrons; both to other Italian-Americans, as well as a growing number of other ethnicities as well.

This, in turn, worked well with an increased demand for ready-made food, required due to an increasing percentage of women joining the workforce and thus unable to cook at home.

Now, this part was easy to answer, but why PIZZA of all the possible Italian dishes?

There is no definite answer, as usual for matters of taste and preference. I would approach it from a perspective on not a historian, but an archaeologist of technology.

Pizza, at its most basic, is simply a flat-bread with cheap toppings. Flat bread is by far the most popular dish in the world, eaten in various forms by virtually every culture known to historians, but especially popular among Mediterranean cultures. It is very cheap, calories-dense, does not spoil easily, easy to transport, share and cut, and can be eaten with anything.
We know it as pita, naan, flatcakes, or chatapouri, but its pizza that manages to be both cheap, and "luxurious" looking enough to be sellable to strangers who might be not familiar with it. It requires no rare ingredients, the procedure of making it is easy, and about the only non standard piece of equipment needed is a wide oven. It is also easily stackable and transportable, which was a non trivial matter to delivery men in 1950s automobiles and bike delivery.

Candeloro, Dominic. "Suburban Italians" in; Ethnic Chicago

Fischer, Claude America calling: A social history of the telephone

Liz Barrett , Pizza, A Slice of American History"

Turim, Gayle. "A Slice of History: Pizza Through the Ages

This requires agreement that Pizza is indeed the "primary" delivery food. I'm really not sure that it is outside certain cultures and I think we'd have to frame the question in recent US history to make that premise true.

Historically we know that Roman towns would often have numerous thermopolia, places that served hot food, and there's no reason to think that food wasn't being delivered to customers from them. Indeed it would be harder to think of reasons why food wouldn't be delivered from them. You might say "Aha! Pizza!" at this point but unfortunately it seems that what we Moderns think of as a pizza probably wasn't something you could buy 2,000 years ago. Maybe even 200 years ago (Mattozi 2015).

We know that within those last 200 years the number of food deliveries has burgeoned across the world. On the Indian subcontinent the system of dabawallahs, fully organised since the end of the 19th century, delivery an eye-watering number of meal tins per day (Roncaglia 2013). We know that Korea has a history of noodle delivery dating back to the mid-1700s (Hwang Yun-seok c. 1768), something which became widespread during the 19th century (Dae Young Kwon, 2019)

So although we can't definitively pin the first food deliveries on the Romans we know that widespread food deliveries were happening long long before a pizza that we'd recognise existed (Alcock 2006), we know that long before Pizza delivery became common in the USA food delivery was widespread in India and Korea (and likely many more places), and we can be pretty sure that the majority of those foodstuffs weren't pizza - although Korea has embraced Western food quite robustly in the last 40 years.

All this leads to an entirely different question: what makes you think that "pizza had the entire delivery market locked down"?

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Alcock JP, 2006 Food in the Ancient World

Dae Young Kwon, 2019, Diet in Korea

Mattozzi A, 2015, Inventing the Pizzeria: A History of Pizza Making in Naples

Roncaglia S, 2013, Feeding the City: Work and Food Culture of the Mumbai Dabbawalas

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Edited to remove reference to another answer which has since been removed