I see a difference in the economic philosophy that the two approaches took. The progressives held to a basic conservative approach to economic management: gold standard, balanced budgets & low peacetime taxes. They were focused trying to soften the adverse effects of capitalism, not interfere with its working. This is a conservative approach towards economics. In social policy they were more progressive in the modern sense: pro women's suffrage, prohibition, consumer protection, pro-welfare.
The New Deals moved to the left on economics. They embraced the populists pro-inflation/anti-banker stance by taking the country off the gold standard and regulating Wall Street. ( In comparison the progressives put in the Federal Reserve which put government economic management in the hands of elite bankers). FDR reigned in the Fed by putting a New Dealer in charge, and during the 1940's the Fed printed money as needed to maintain low interest rates. This is a recipe for inflation so he simply made it illegal to raise prices/wages without permission. For wages the system was gamed. Companies were free to raise wages to about $11/hr in today's money. (This they did because with wartime stimulus workers were hard to find). If you were in the middle-tier of income you had to get permission from the local office, who would be somewhat responsive to local issues. These folks saw their wages keep pace with inflation. For wages in the upper tier you had to get permission from the Washington office who just said no. The results were a compression in the spread of incomes, force economic inequality amongst income-earners. For the rich they had a 91% tax rate on dividends, interest, salaries and bonus income (where most of their income came from). Relative to the rest, the rich became poorer, a trend that had started during the Depression, was intensified during WW II (by the policy above) and continued at a slower rate for a couple of decades afterward. The New Dealers also made labor actions full legal and favored labor over management in disputes.
Yes, their goal was the save capitalism from itself, but the medicine was harsh and bitterly resented as shown by the Conservative movement that began in the 1950's, blossomed in the 1970's and triumphed in the 1980's and since.
Yes they were conservative (small c) but they were conservatives of the left in that they took some of the beliefs with informed their policy thinking from the economic left. They were not radicals of the left (like you), who seek to change the economic
roots (or fundamentals) of society (radical means root).