I'm going to address the question of the relationship between American living standards in the 1920s, the stock market crash, the Great Depression and the New Deal. Hopefully someone else will be able to answer the aspect of the question about FDR's plans. 

I am interested in where you got your description of the 1920s as "the average American during that time was rural, poorly educated, highly likely to die from any number of diseases at a young age, and lived a life of penury and dire poverty."

From US Census data, in 1920 the urbanisation rate was 51%, rising to 56% by 1930, so the median American in the 1920s was urban, not rural. Life expectancy at birth rose from 53 years in 1920 to 59 in 1930. Illiteracy rates fell from 6% to 4%. And the OECD estimates that real wages, even of unskilled labour, in the USA were amongst the highest in the world in the 1920s, possibly even the highest (de Zwart, 2014, Table 4.6, page 11). Obviously compared to modern day living standards in Western countries, most Americans in the 1920s were poorer, less educated, and at greater risk of dying of disease as there were no antibiotics and a lot fewer vaccines than we have these days, but the picture of the average American as living "a life of penury and dire poverty" is very misleading in the context of economic development. 

The causes of the Great Depression are highly debated by the economics profession, the Economic History Association's website has a good article by Randall Parker discussing the sequence of events and the various theories as to why. The formal start of the Great Depression in the USA is in August 1929, so before the stock market crash in October 1929, but there are some theories linking the crash to the size of the Depression, to quote from the eh.net article:

Several authors have offered explanations for the linkage between the crash and the recession of 1929–30. Mishkin (1978) argues that the crash and an increase in liabilities led to a deterioration in households’ balance sheets. The reduced liquidity2 led consumers to defer consumption of durable goods and housing and thus contributed to a fall in consumption. Temin (1976) suggests that the fall in stock prices had a negative wealth effect on consumption, but attributes only a minor role to this given that stocks were not a large fraction of total wealth; the stock market in 1929, although falling dramatically, remained above the value it had achieved in early 1928, and the propensity to consume from wealth was small during this period. Romer (1990) provides evidence suggesting that if the stock market were thought to be a predictor of future economic activity, then the crash can rightly be viewed as a source of increased consumer uncertainty that depressed spending on consumer durables and accelerated the decline that had begun in August 1929. Flacco and Parker (1992) confirm Romer’s findings using different data and alternative estimation techniques.

Mainly modern theories about the causes of the Great Depression focus on monetary and financial mechanisms. This is in line with Milton Friedman and Anna Schwartz's 1963 book A Monetary History of the United States, 1867–1960, that showed a drastic fall in the money supply in the US in the 1930s and subsequent academic work on a broader international scale relating different countries' recoveries to when they left the Gold Standard (or in Spain's case, weren't on it in the first place). 

The New Deal was a complex and changing set of policies that makes it extremely hard to assess as a whole, apparently FDR's idea was just to try everything and see what worked. In 1995 (a few years ago now), a survey was done of economists and historians who were members of the Economic History Association. In this survey, they were asked to agree or disagree with a number of statements about US economic history, including the statement "Taken as a whole, government policies of the New Deal served to lengthen and deepen the Great Depression", found that 27% of economists agreed, and 22% agreed with provisos, while only 6% of historians agreed, and 21% agreed with provisos. (Whaples, 1995). This was one of the largest areas of disagreement in the whole survey. Unfortunately we don't know whether those who disagreed thought that the New Deal improved matters or just that it was irrelevant.

In terms of the social security aspects of the New Deal, which were aimed more at protecting the poor and vulnerable, rather than at directly improving aggregate economic outcomes, I note that systems of social security were fairly widespread amongst rich countries at this time. The New Deal did however fail to include measures for improving the economic situation of black Americans by tackling racial segregation or Jim Crow laws in the South, it didn't even desegregate the US Federal Service. 

In terms of comparison to the modern day situation, I can't comment. Not only because today's situation is well within the 20 years rule, but because we don't yet know how long economic activity is going to be shut down for.  

Sources

Parker, Randall. “An Overview of the Great Depression”. EH.Net Encyclopedia, edited by Robert Whaples. March 16, 2008. URL http://eh.net/encyclopedia/an-overview-of-the-great-depression/

Robert Whaples (1995), Where Is There Consensus Among American Economic Historians? The Results of a Survey on Forty Propositions, The Journal of Economic History, Vol. 55, No. 1 (Mar., 1995), pp. 139-154 (https://www.jstor.org/stable/pdf/2123771.pdf)

de Zwart, Pim, Bas van Leeuwen and Jieli van Leeuwen-Li (2014), “Real wages since 1820”, in Jan Luiten van Zanden, et al. (eds.), How Was Life?: Global Well-being since 1820, OECD Publishing. http://dx.doi.org/10.1787/9789264214262-8-en