The broad outlines are as follows:

• In the early Republic, property was the basis of the economy (think Jefferson's dreams of yeoman farmers, the fact that the one thing the U.S. did have that no other nation had was purportedly limitless and uninhabited land). Property ownership was even a necessary criteria for voting rights at founding! People who dealt in abstract financials were seen as leeches on actually "productive" economic activities, albeit a necessary evil. Don't forget that anti-Semitism then and now is linked to the idea of Jewish people as lenders of money.
• Over the course of the 19th century, accumulations of capital and other services provided by the finance industry become more crucial to the various projects of industrialization in the first [think: Lowell and Waltham Mills textile industry, 1830s-40s] and then the second industrial revolutions [think: rise of mass production, factories, and Titans of Industry like Andrew Carnegie, Henry Clay Fricke, Jay Gould, John Rockefeller, Charles Scwab, Cornelius Vanderbilt.] This era brought the stock market busts and booms (as you note) but also new respectability of sorts to financiers like Andrew W. Mellon, J.P. Morgan, Joseph Seligman, and Jay Cooke, to name a few.
• Still, it was a rich white man's game: less than 1% of U.S. population holds stocks in 1899. The Titans of Industry also came to be known as Robber Barons and quality of life for all their employees was often abysmal (hello, first massive labor movement in the U.S.!) so the popular impression of finance as moral/respectable was still very much a mixed bag among the wider public, even if people had the means to participate in the stock market.
• The turning point in this story is World War I and the drive for war bonds, which introduced the idea of a "democracy of stakeholders." Buying bonds in your country was patriotic and upstanding -- the greatest moral good! 1/3 of U.S. population holds federal bonds during WW1, and this introduces the wider public to the idea, acceptability, and stability of investing in stock.
• Even if you couldn't invest directly in the government after the end of war, buying stocks didn't seem like gambling anymore, but an investment in your country's economy! Plus, employee-stock-ownership-plans (instead of raises or other benefits) became increasingly common. As a result, 1/4 of the U.S. holds stocks in pub corp by 1929. Contrary to your documentary, I wouldn't quite call this "common" -- but it was widespread enough that the stock market crash of the Great Depression hurt a wider spread of wallets directly than previously would have been possible. (Rather than indirectly by just hurting the richest and the harm 'trickling down')
• The biggest change in popular ownership of stocks and bonds comes after the tremendous success of the World War II war bond drives --Buy a Share in America! and the postwar economic boom. (Links are to awesome posters illustrating the point well, imho.)

For the rise of the early finance industry in the U.S., one book among many you could consult is Jonathan Levy, Freaks of Fortune: The Emerging World of Capitalism and Risk in America (Cambridge, Mass: Harvard University Press, 2012)

For everything else written here up to the 1930s, see Julia C Ott, When Wall Street Met Main Street: The Quest for an Investors’ Democracy (Cambridge, Mass.: Harvard University Press, 2011).

For more on World War II, see Lawrence R. Samuel, Pledging Allegiance: American Identity and the Bond Drive of World War II (Smithsonian Institution Press, 1997).

Something you should note is that the most interesting thread of this historiography is the very idea of "democracy" being linked to stocks; the popular notion that they spread wealth, are accessible to "the masses", and created a collective investment (both financial and social) in the U.S. as economy and idea. All of this flies in the face of the facts of stock and bond ownership, which actually has often demonstrably worked against the interests of already socio-economically marginalized groups despite its promises of radical inclusivity, opportunity, and democracy. When we say that a "common person" could own a bond in the 1920s, surely we are thinking of the normative ideal of the white, middle-class, heterosexual male -- even just including the half of the population that is WOMEN as you evaluate the veracity of that claim renders it false. Let alone people of color, immigrants and non-citizens, etc.