I'm going to focus on the question of whether the US transitioned from a boom-and-bust cycle to a more stable one post WWII, and if so can this transition likely be attributed to FDR and/or WWII.

First up, a disclaimer, it is hard to rigorously attribute a change in economic outcomes in a particular economy to any particular cause. There are numerous things changing all the time in any economy of any size. Quite often an explanation that sounds really good and convincing for one time period fails to hold up in another time, or fails to hold up across countries. So, as you say, there's no definite consensus amongst economic historians on these issues. 

It used to be commonly accepted that the US saw much less volatility in economic output post WWII than it did before WWI. The inter-war period was of course highly volatile due to the 1920s boom and then the Great Depression. However this consensus was questioned by Christina Romer in a couple of papers published in the 1980s. Romer argued that the decline in volatility was due to better data measurement with the establishment of modern statistical collections (from the 1930s into the 1950s). In an updated article in 1999, Romer revisited her work, and estimated that US volatility of industrial produciton fell from 6.2% from 1886-1916 to 5.0% from 1948-1997, and NP from 3.0% to 2.5% (table 1).   This is a reduction in volatility, but not a drastic one. And Romer states that it's not a matter of a change from large recessions pre-war, to large booms post-war, instead the distribution of annual changes were similar, with the post war one being slightly compressed. 

Romer's work is somewhat controversial, obviously one's results for this depends in part on how figures for the pre-war era are constructed. As another view on this issue, Backus and Kehoe (1992) looked at business cycles for ten countries for which they had a century of relevant data: Australia, Canada, Denmark, Germany, Italy, Japan, Norway, Sweden, the United Kingdom, and the United States, several of these countries having better raw data sources pre-WWI than the US (e.g. the UK had an annual income tax in effect from 1842, the Australian government played a larger role in the Australian economy in the 19th century and collected data, etc). They found that in 6 of the 10 countries studied (Denmark, Germany, Italy, Japan, Norway and the United Kingdom) pre-war fluctuations were no more than 60% larger than those of the postwar era. Australia, Canada, Sweden and the United Stats saw larger fluctuations pre-war than post, with US output fluctuations about 1.9 times higher pre-WWI than post WWII. 

So it does seem likely that there was an increase in stability of output post WWII in the USA, but it's not entirely clear, it's not like it was a complete structural change, and it's not unique to the USA. 

There have also been two significant changes in the US economy over the 20th century: a rise in government spending (particularly in WWII) and a shift to services making up a larger share of the economy, both of which should reduce the volatility of output. The 20th century US government can easily borrow and keep spending if tax revenues fall short, and services can't be stored so aren't as volatile as manufacturing - a service-based company can't run down inventories, nor can a consumer necessarily stretch out some legal advice for another year like they might a machine (and growth in government output, apart from defence, tends to be mainly in services too). Note: the increase in government spending and in services have happened in other developed economies. 

There are also other possible causes of the increase in stability apart from FDR. The US Federal Reserve was established in 1914, meaning that there was a central institution able to effect large changes in short-term interest rates (famously Milton Friedman and Rose Schwartz in 1963 argued that the Great Depression was mainly the result of the US Fed stuffing up money policy (unintentionally).) The 1930s and 1940s also saw the increasing influence of John Maynard Keynes, a British economist, who argued for more active government policy to manage macro-volatilty, and these ideas continued to be influentional after his death, into the 1950s (and indeed today). Romer does credit deposit insurance, a specifically 1930s phenonomen, for reducing bank runs and financial panics, but this is one cause amongst many. 

Romer notes that the 1980s and 1990s were much more stable than the 1950s-1970s. Inflation has been under control, and central banks have avoided the policy mistakes of the 1930s, though she ends her 1999 article on the cautious note that:

Does this mean that there really is a “new economy” today where cycles are largely nonexistent? Perhaps. We have had the potential for greater stability since the advent of aggregate demand policy, automatic stabilizers, and deposit insurance. In the last decade or so we have used these tools to counteract shocks and have not created new ones. As long as policy continues to make few mistakes toward overexpansion, policy induced recessions to control inflation should continue to be rare.

But describing the source of the change in this way makes it clear that the trend toward greater stability could be quickly reversed.

(And due to the 20 year rule I will not comment on volatility in the 21st century).

So in summary, the US economy does appear to have become more stable post-WWII than it was pre-WWI, but the change wasn't that drastic, nor unique to the USA. And plausible drivers of the change include things outside FDR's presidency, namely the establishment of the US Fed in 1913, higher government spending post WWII, a general structural change to services playing a larger role in the economy, and improved monetary policy. All this said, FDR's financial regulations such as deposit insurance likely played a role.   

Sources

David K. Backus and Patrick J. Kehoe International Evidence on the Historical Properties of Business Cycles, The American Economic Review, Vol. 82, No. 4 (Sep., 1992), pp. 864-888, http://users.econ.umn.edu/~pkehoe/papers/BKaer1992.pdf

Romer, Christina, D. 1999. Changes in Business Cycles: Evidence and Explanations. Journal of Economic Perspectives, 13 (2): 23-44. https://www.aeaweb.org/articles?id=10.1257/jep.13.2.23