"What are some problems from 19th century the Engine could solve and thus provide substantual economic boost?"
For those readers who don't know what we are talking about, Babbage's envisioned Analytical Engine would have had about the same capabilities as a typical ordinary programmable calculator, such as the famous HP-42S (of which a free open source re-implementation is available as Free42). Of course the HP-42S was very very very much faster, but speed is not of the essence. But what is of the essence is that the HP-42 was also very very very much cheaper than the Analytical Engine, and it was mass produced.
There are literally thousands of programs available for the HP-42, and any of those could, in principle, be implemented on a functional Analytical Engine.
But the competitors of the Analytical Engine were not mysterious programmable calculators from the future, but human computers. The Analytical Engine was to be a mechanical programmable calculator, with ordinary mechanical speed; certainly not any faster than 100 human computers.
Let's try to understand the economics.
In the 1850, 50 real gold pounds sterling per year would have been a fantastic wage for a human computer; that would be about 7000 debased paper pounds sterling of 2022. Mr. Babbage burned through more than 17,000 pounds sterling (about 2,500,000 British pounds in 2022 money) before Her Majesty's Treasury pulled the plug. With that money the Treasury could have employed a hundred human computers for three years, computing whatever they wanted to be computed. With the advantage that the human computers did not require any huge capital investment, but rather they would have been pay-as-you-go.
The point is that one fantastically expensive, slow-as-molasses Analytical Engine would not have provided the British Empire with any competitive advantage. It could not do anything that a team of human computers could not do, it would not have been any faster, and it would have costed a lot more.
Meanwhile in America
It may be instructive to compare Babbage's visionary but utterly impractical dream with the severely limited but eminently practicable inventions of Herman Hollerith, who concentrated on designing and building electromechanical tabulating machines actually useful in real life.
Hollerith's machines were first used to tabulate the results of the 1890 U.S. Census; while the results of the 1880 census took eight years to tabulate, the results of the 1890 census were completed in six years, although the population had increased by 25%. Hollerith and other inventors continued to develop and refine unit record equipment, which became ubiquitous in commercial and governmental organizations throughout the first half of the 20th century; general-purpose digital computers only replaced them in the late 1960s or early 1970s.
In 1911, Hollerith's Tabulating Machines Company joined four other companies to form the Computing-Tabulating-Recording Company, which, in 1924, changed its name to International Business Machines, better known as IBM.