It just defies all logic that only a hundred years after a global thermonuclear war, there's no sign of that war ever having happened.
After a century, what kind of signs would be expect to remain? The radiation would be gone, the land recovered, survivors would mostly be gone.
There could be monuments in many places. Destroyed buildings, even entire cities, preserved perhaps.
Most of the world's major cities were in ruins
Give the death count and the recovery time this seem unlikely.
and only a short time after that, they're all rebuilt?
With the world population reduced by 600 million, many cities simply would not be immediately rebuilt because there would be
no one to live in them.
Cities (many of them) are where they are for a reason and new communities might eventually reappear, but could be much smaller than the originals.
It would take a hundred years just to rebuild ONE CITY, let alone all of them.
How long to rebuild Hiroshima, fifteen to twenty years? It's bigger now than prior to the nuclear bombing.
And again there wouldn't be a pressing need to rebuild some of the cities.
How can you rebuild when there's nothing left to rebuild with?
The majority of the population were not killed, so there could be international aid from the outside for the survivors.
Recovery might not consist of returning everything to the state it was before the war. Recovery could mean helping survivors, rebuilding the international economy and trade, dealing with the repercussions outside the war zone. If the combatants were China and India (one possibility) then the world would have to realign without them as major players anymore.
I’m saying that it doesn’t jibe with how a Third World War is feared to go
I guess that comes down to what your preconceived idea of what a third world war is supposed to be. Nothing says it has to have been a massive nuclear exchange between America and Russia with tens of thousands of warheads.
Other than something vaguely called the Eastern Coalition we have no idea who the combatants were, America would seem to not have been one of the combatants.