We are mostly arguing semantics. I have no problem if you want to say most of the higher education came post-war, but I would point out that was just a planned continuation of the New Deal. I don't know of any government agency that has profitability as a goal - the TVA's purpose was to electrify rural areas and provide cheaper electricity & modernize agriculture, which it did.
The wage compression that created the middle class was unmistakably during the New Deal. Economists typically attribute this to wage controls and unionization. You aren't going to find a pension program with the security or anything near the low administration costs of Social Security. Even today, most elderly Americans would be in poverty without it.
True, the widespread gains in health were probably motivated by the war. As I recall, 60% of recruits were rejected in many areas because they had STDs or were missing too many teeth. Nevertheless, it took massive Government programs to turn it around.
According to the OECD, the large investment in science & engineering in the mid-late 40s is responsible for the US' early lead in productivity and education. I know you don't attribute this to the New Deal, but it doesn't really matter because it was still the Government's demand-side investment that created these gains.
As I said before, the early New Deal was rather small - especially by the standards of Government today. However, the Keynesian policies of the New Deal DID pull us out of the Depression and created a modern society unseen even in the best years of the gilded age. WW2 was just a magnification of these policies.