realized that fall occurs exactly when the sun sets in the west

Mmmm, that's not exactly true.

Autumn starts at the fall equinox, i.e. when the center of the sun (moving along the ecliptic) crosses the celestial equator (downward as seen from the Northern hemisphere).

That can happen at any odd moment day or night, not necessarily the moment when the sun rises or sets.

Suppose the sun crosses the equator just after midnight, then at the moment of sunset she has already travelled a little over 44 arcminutes (3/4 of a degree) along her path and the center of the sun will thus not be exactly due west when she sets.

As the angular size of the sun as seen from the earth is a little bit over half a degree, the full disk of the sun "misses" due west totally.