I'm generally skeptical of attempts to explain the economic development of a single country, or a single brief time period. There's so much going on in any large economy at a single point of time, large amounts of which we have only the crudest measures even with all our modern statistical collections that I suspect mostly people are seeing what they already believed - we're interpreting ink blots. (I'm certainly not immune to this bias.)

There are some exceptions to my skepticism, there are ways of destroying a country's economy, such as a long-running civil war or hyperinflation. And if a country is highly dependent on one or a handful of products then its easier to understand - oil is about 40% of Saudi Arabia's GDP, so oil price change * oil volume change explains a lot of changes in the Saudian economy. 

But for more diverse economies, every simple rule proposed seems to have an exception. Germany managed to industrialise without an empire. Japan managed to industrialise without being full of Europeans. Switzerland managed to industrialise while being land-locked. Singapore thrived as an independent state. Ireland, meanwhile stayed poor for decades post independence in 1921, despite an educated population, electrification and being an island. Then Ireland took off in the 1980s. Economic growth is really about people seeing more and more valuable (to themselves) ways to use resources (I mean, really, that's at the core of GDP), and that sort of human ingenuity can operate in a wide variety of circumstances. 

Economic history of the 18th and 19th centuries also tends to involve studying trade. This is because customs duties was an important source of government revenue so there's significant statistics on imports and exports (though surviving data is incomplete and of course people sought to avoid duties so the numbers aren't fully reliable). This creates a bias in what stories are told, and can be told.  The economic historian Deidre McCloskey argues that historians and policy makers alike tend to place too much importance on trade and the main causes of growth for large economies are domestic (large economies: obviously Luxembourg is a different story!). 

Another factor is that of opportunity cost: if the USA hadn't had cotton and the cotton gin presumably it would have found some use for most of the resources that went into cotton. Presumably a bit less profitable use, but that means that we can't just measure the contribution of cotton by measuring the associated flows, we need to estimate the next-best alternative. This is an important issue in economic history, at least ever since the work of Robert Fogel in 1962, who set out to measure the contribution of railways to American economic growth and wound up concluding that:

that the level of [American] per capita income achieved by January 1, 1890 would have been reached by March 31, 1890, if railroads had never been invented. (Quote from Lance Davis's review.)

Basically railways could almost entirely be replaced by canals and better roads. Robert Fogel's work sparked a lot of debate and criticisms but the broad picture of relative un-importance of railways remains standing. 

A third factor is that these debates tend to be very Anglo-centric: what was the USA and the UK doing? Sometimes France is mentioned. But the Atlantic slave trade affected Africa too. I'm no expert on any African country's history but I understand that the African slave trade not just affected the people directly taken and ripped from their families but also the Africans who remained. Firstly every human taken from Africa to labour in American cotton fields was a human taken from the African labour force. And, given the death rates of the Atlantic crossing the numbers of Africans taken were higher than that. Beyond the direct costs of fewer humans, African peoples naturally struggled valiantly to avoid enslavement. However the role of Africans in slave-raiding parties meant that this was not just a struggle against Europeans and Americans but also an internal intra-African struggle.  There's been a strand of economic history research into the impact of the slave trade on Africa, which tends to the conclusion that the slave trade harmed West Africa's economic development by harming social capital and political functioning and that these effects are still visible today. It is better to have a richer, more productive trading partner than otherwise, the gains to the northern states and Western Europe from US cotton needs to be offset by the losses from African potential production to form a view about whether US cotton and slavery was an overall positive or negative to industrial capitalism. But a question like "how much would Africa have exported in the absence of slavery" is a counter-factual beyond the tools of economic history.

In short, it is pretty implausible that any single commodity or technology or institution was essential to the economic development of industrial capitalism. 

On the other hand, the argument that slavery affected and continued to affect American politics, as well as African politics, is a much stronger argument and the USA has been a major global player since the 20th century so to some extent there's a route there for slavery to affect world development. On the third hand I'm also a skeptic about the ability of American politicians to influence what happens in economic development, but that's an argument for another day. 

  

Sources

This answer mostly draws on years of reading. The two specific sources I used are:

Davis, Lance, EconomicHistory.net Review of Robert Fogel's Railroads and American Economic Growth

Whatley, Warren & Gillezeau, Rob. (2011). The Impact of the Slave Trade on African Economies. 10.11126/stanford/9780804771856.003.0004.

For some more general skepticism about our ability to explain economic developments, see

McCloskey, Deidre, 2010, Bourgeois Dignity: Why Economics Can’t Explain the Modern World,, and much of her work at http://www.deirdremccloskey.com/articles/index.php (her theory that it was actually about ethics and rhetoric is more dicey, but she's good at critiquing other theories). 

Also 

Easterly, William, 2006, The White Man's Burden: Why the West's Efforts to Aid the Rest Have Done So Much Ill and So Little Good, Penguin Publishing Group