It's mostly increases in targeting and accuracy -- in the era of air-dropped bombs, and early ICBMs and cruise missiles, having a reasonable probability of kill

It's mostly increases in targeting and accuracy -- in the era of air-dropped bombs, and early ICBMs and cruise missiles, having a reasonable probability of kill on a hardened target required an absolutely enormous warhead. With the minuscule circular error probables of modern weapons, the "city killers" of the 1950s and '60s simply aren't necessary in modern nuclear doctrine. As you mentioned, fallout is greatly decreased by the reduction in warhead size. Additionally, as airbursts are more effective at causing damage, they have taken precedence over ground bursts, and have a substantial impact on the reduction in fallout. The concept of a nuclear winter is primarily related to the expected firestorms in cities and forests, which would throw colossal amounts of soot into the air. With infrastructure devastated by the strategic weapons exchange, the resources to effectively fight the fires weren't expected to be in place, so it was thought they'd burn out of control. While there has been debate over how realistic it is that a nuclear winter would occur from a strategic nuclear exchange, nobody doubts that it would be catastrophic.