I think it may be too optimistic to declare the USA immune to dictatorship, but the path to it would look nothing like the quasi-legal route the OP suggests. A law to mandate one party only? That's how it might consolidate itself, but first there would be violence without recourse to law, except insofar as we might say the unilateral claim of some leader to hold emergency governing powers is a "law." So it would be if the dictator stayed in power of course!
I don't think FDR wanted to be a dictator on lines of Hitler, Stalin or Mussolini--this does not mean he would never bend rules to stay in charge. I think there might have been circumstances in which he assumed emergency powers, and would not subject them to Congressional review until after the emergency, and if in his view being held accountable in that forum would undermine a necessary victory, he might suppress it. But only if he felt the alternative was far worse! OTL he governed with the Republicans dominating Congress sometimes, and relied on fair elections to remain in office--note that the court packing scheme was a threat, not a resolved plan he was prevented from carrying out, and the threat had the effect of persuading a swing SCOTUS justice to switch sides and form a new court majority not devoted to checking everything he did.
Dictatorship may have happened at the hands of a reactionary faction panicking at the dangers of the discontented mobs of the Depression years. It may have taken the form instead of such radicals deciding the regular political system was too slow to address their urgent needs, especially if the system didn't throw up someone with the combination of credibility, energy and gravitas FDR offered but put up some nonentity instead. OTL Huey Long of Louisiana wanted to play a long game, trying to throw the '36 Presidential election to some Republican so that the people would be in a mood to back his more radical and sweeping "Share the Wealth" program in 1940, so he isn't a good candidate for a dictator in a hurry. Perhaps if the Democrats had failed miserably to offer a promising candidate in the 1932 election though maybe Long would respond to mass discontent by offering to lead an emergency regime.
The thing is, it would happen by people stepping out of line and breaking the normal rules. My working definition of a revolution is when a decisive majority of the military decides they should be backing some rebellion instead of suppressing it. Modern stereotypes about the mentality of military people, even officers, won't serve us too well here; the officers tended to lean conservative but there are plenty of examples overseas of leftist military coups and some examples of very popular military leaders with populist, if not necessarily, progressive ideas. Dictatorship would happen via civil war, because even if one faction appeals more strongly to the majority of soldiers and sailors, the other side will have its adherents and they would fight each other in the streets, along with militias partisan to both sides.
The civil war might drag on and result in all around ruin, and the nation, exhausted, might submit to some gray nonentity just to have peace. Or one side or the other decisively wins early on and calls the shots thereafter.
After someone has won, they might gut or hamstring democracy to keep in power and maintain stability. More likely they'd pretend to be banning all parties for the sake of unity; the handpicked "unity" leaders would in fact be a new party of course. There's no way that one party, even if it dominates a legislature, can simply vote to perpetuate itself and ban its rivals without the other side calling foul and getting a lot of public sympathy and support. The attempt to do this might cause the civil war that settles matters, but it cannot accomplish its aim just by a few hundred legislators saying "aye" and ignoring the "nays."