The economic literature is unclear whether patenting is a hindrance or a benefit to the early states of industrialisation. The strongest arguments for it being a benefit are indirect - via would-be inventors being inspired by the hope of winning a large award, or via cultural formation effects. 

It's very difficult to reliably attribute a particular country's economic development to particular factors because economies are made up of complex interrelationships, and people are fairly smart about adapting to their particular circumstances. So just because an economy developed one way doesn't mean that it had to develop that way, under different circumstances people might develop a different economy, but still a prosperous one. E.g. a lack of sea-access did not stop Switzerland from becoming a rich country. 

For questions like these, therefore, economists typically look across countries for correlations between various aspects of each country and economic development, informed by theory (and said empirical data then informs theory.) Unfortunately, for periods before about the mid-20th century, when the modern national statistics systems started to come into operation, our economic data is mainly scanty and unreliable, thus making cross-country studies very limited at best.

Theory

The theoretical literature on IP is ambiguous. Roughly, patent theorising concludes that patents directly creates an incentive for innovation, but indirectly raises the costs of innovations that combine new ideas and inventions, and can create short-term monopolies that may become long-term ones (path dependency). So who knows?  

Patenting and economic growth: empirics

Empirically: introducing or strengthening IP protections is associated with an increase in patenting and the use of patents. It's much less clear that this results in an increase in innovative activity overall, stronger IP laws may change what gets innovated. Some have found that strength of IP rights is associated with more investment and R&D investment, but the relationship appears to be depend on the existing economic development (eg high-income versus low-income) of the countries in question. There are some problems with this work though:

• This empirical work can't control for how the economic development of a country might affect its willingness and ability to increase patenting protections. 

• Less-developed countries tend to have fewer and lower-quality statistics, casting doubts on any conclusions using their statistics (in either way). 

• This empirical work by its nature focuses on the 20th century, when sufficiently detailed data sets become available. 

There are some comparisons from the 18th and 19th centuries. For a start, during the start of the Industrial Revolution in Britain, up until 1852, patenting was very expensive - there was a filing fee of 100 pounds for England and 350 for the UK as a whole, not counting time spent. And enforcing the patent was even more expensive, with judges hostile to it.  A study of the British exhibits at the Crystal Palace exhibition in London in 1851 found that only 11% of them were patented, and, of the subset of those exhibits that won awards, less than 16% were patented. Of course some Brits did take out patents, e.g. James Watt (whose patent on the steam engine was famously extended by parliament), and arguably the large fortunes reaped by a few inventors may have motivated a lot more people to make inventions in the hope of basically winning a monopoly. The 1852 reforms, lowering the price, led to a sharp increase in patents, but obviously the industrial revolution was well underway by then.  

The Netherlands also had a patenting system from the 16th century, and fairly high incomes, but it didn't industrialise (though Dutch incomes remained strong) and Dutch patents filing dropped significantly from the 1670s. Joel Mokyr, in 2009 (who most of this is taken from) argues that the main contribution of patents to economic growth in Britain was via the example of a few lucky inventors who got rich from their patents, setting up other people to try basically a lottery (like multi-millionaire NBA players inspiring millions of kids to practice basketball fanatically). 

In a similar vein, Laura Ford, in 2017, published an interesting paper arguing that intellectual property law, in the USA, contributed to the formation of a culture (Ford specifies a 'political culture') based around a belief in new technologies and new cultural products, as a means for achieving economic growth. Therefore more invention and creativity took place, thus forming not just new inventions but social capital that led to industrialisation - through mechanisms such as patent-licensing creating formal links between mills (in Fod's case study, the Boston Manufacturing Company (Waltham) and the Lowell manufacturing companies were the centers, providing not just formal patenting licences but skilled mechanics to other New England mills, along with other support such as managerial techniques). So it wasn't so much the protection of the patents themselves that was valuable for US economic growth, but the social capital built up via their impact.  

Unfortunately neither Mokyr nor Ford provide any information that would allow us to evaluate the relative importance of patenting, both to the cultural impacts, and to economic growth (and, in their defence, I can't think of a method by which they could have quantified this given the limited data available from the 18th/19th centuries). 

So, it's possible that patents contributed towards US economic success, there are mechanisms by how it could. But I don't know of any academic source in economic history that argues that patents are definitely a huge source of economic growth. 

Sources

Mokyr, Joel. 2009. "Intellectual Property Rights, the Industrial Revolution, and the Beginnings of Modern Economic Growth." American Economic Review, 99 (2): 349-55.

Bronwyn H Hall, Patents and Public Policy, Oxford Review of Economic Policy, Volume 23, Issue 4, 1 December 2007, Pages 568–587,

Albert G.Z. Hu  I.P.L. Png, Patent rights and economic growth: evidence from cross-country panels of manufacturing industries, Oxford Economic Papers, Volume 65, Issue 3, 1 July 2013, Pages 675–698, https://doi.org/10.1093/oep/gpt011 

Natália Barbosa, Ana Paula Faria, Innovation across Europe: How important are institutional differences? Research Policy, Volume 40, Issue 9, November 2011, Pages 1157-1169

Ford, L. (2017). Intellectual property and industrialization: legalizing hope in economic growth. Theory & Society, 46(1), 57–93. https://doi.org/10.1007/s11186-017-9285-3

(Note a chunk of this comment came from an earlier comment I wrote in answering a question about whether the USA flourished because it didn't enforce IP laws. Sadly I can't find the link.)