I dispute his premise that the U.S. experienced a conservative backlash. Yes, the U.S. gradually became more conservative in the 1960's and 1970's than it had been in the 1940's, but I would argue that this was likely to occur anyway, and that the New Deal Era was an anomaly.
Essentially, I think that the U.S. is more conservative of a country than most of those in the industrialized world. This has almost always been the case in U.S. history, especially after the 1850's.
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Yes, AFAIK the New Deal was something of an anomaly. Without having studied US recent history in much detail, I do get the impression that FDR really had to fight to get the New Deal policies enacted, against opposition from his own party as well as the Republicans.
As to whether the US becomes more conservative or more liberal in response to a Nazi Cold War:
It seems likely that the US would want to disassociate itself from anything considered to be "Nazi-like". The issue here is how such things would be defined. There are some issues which would seem to be obvious points of conflict, for example the various US eugenics programmes. Before anyone says "How dare you! US? Eugenics? That was them!" etc, I'd like to point out that the US did have
eugenic sterilisation programmes, in various states, for the greater part of the 20th century.
Anyway, my point is that eugenics in the US wasn't suddenly abandoned after OTL WWII, it went into more of a gradual decline: so why would it be suddenly abandoned in an anti-Nazi Cold War?
If we instead consider economics: with the New Deal in place, the US I think moved more towards a planned economy. It didn't become one in full, but it moved in that direction. Now, whether in a US-Nazi Cold War scenario, the US decides to move away from this, is rather unclear. When in the OTL Cold War against perhaps the ultimate statists, the Soviet Union, did the US give up on New Deal programmes because they were examples of planning, and therefore "commie" ideas? Not AFAIK. Many of the New Deal agencies are still around, aren't they? So, even if the Nazis came to be perceived as statists, I'm not sure that the US would automatically give up on economic controls, just because the Nazis do it.
I could keep arguing in this way for a while, but I won't.