You seem to be arguing that FDR was a leftist because of the policies he implemented. But I think what you are missing is why he implemented those policies. I think the truth is he didn’t really have the public interest at heart. His agenda was to contain a growing threat to capitalism in the form of the Communist Party of the 1930s. His strategy to contain the CP was to neuter the party by bringing it into the Democratic party fold, alienating their most militant members, and slowly squashing their agenda. Of course he had to appeal to their interests to do so. But it was a temporary strategy, not a real shift in US policy. There are a few articles on the topic if you are genuinely interested. Here’s one. And here’s a quote from another.
The New Deal reforms Sanders evokes were not the product of a farsighted, enlightened reformer, but responses to tumultuous class struggles in the early and mid-1930s. These reforms sought to contain explosive social struggles and were never truly universal, excluding women and African-Americans, for example. After mass struggle ebbed, Roosevelt shifted back to his original goal of stabilizing US capitalism while moving toward establishing US global domination during World War II. Progressive reforms came to an abrupt halt in the late 1930s, allowing the rollback of many popular gains during the 1940s.
There are two kinds of true, and thus two kinds of ‘real’. There’s the kind where what you believe matters and the kind where it doesn’t. Gravity is the second kind. Step off the top of a building and it doesn’t matter if you believe in gravity, you’re going to fall. Politics is the first kind. If everyone believes I’m the king of North America, well, then that’s the truth. It’s reality. Likewise if everyone in the government believes that Elon Musk can fire whoever he wants, then he can, because everyone will just go along with it.