Any high technology has incredibly long dependency chains. I think you seriously underestimate the difficulty of bootstrapping, say, WW2 level tech from irradiated wastelands after major nuclear exchanges.
In any case, I don't share your low opinion of future humans. They'll be as capable as we are; maybe far more capable.
You also over-estimate and over-weight how destructive nuclear exchanges really are, the readiness state of the world's nuclear arsenals, and the willingness of their possessors to lash out at effectively unaligned countries like Argentina, Chile, Austria, Morocco, Fiji, and I could go on all day.
I don't think you understand the massive difference in scale between detonating a couple of atom bombs vs. thousands of thermonuclear devices, each with at least an order of magnitude (~16 kt vs hundreds) more destructive power. Nevermind the vast fallout dispersal that would blanket the northern hemisphere at least, as well as the ridiculous amounts of soot in the atmosphere from the resulting firestorms that would, to put it mildly, be a bit of a setback for agricultural yields for a damn long time. You might be okay in those unaligned places, sure, bit if you're in roughly half of the world you're pretty much effed.