Another is if it pushes the heat toward areas of weaker insulation and away from areas of stronger insulation. If the outside is cooler than the inside, then the temperature will leak out more rapidly. If the outside is warmer than the inside, then the warmth will leak in more slowly.
You're exactly wrong, and I'll try to show you why:
Consider your two premises for the "hot outside" situation:
(a): There are (at least two) areas with different insulation, and
(b): The outside is warmer than the inside.
If the air is still, you will reach an equilibrium, where the weak insulation spot can't easily cause more heat transfer to the air on the inside, because the air on the inside is already pretty hot. You end up with a room with areas that are hotter than others, and heat transfer within the room is mainly radiation and not convection.
Now, if you circulate the air on the inside, you provide cooler air to the (a) areas. Cooler air which can absorb more heat from the outside. You've disturbed the equilibrium, and go towards a new one where all the air in the room is subject to being heated through the (a) spots.
In effect, you create a convection oven, which works precisely on the principle of bringing cooler air to the heating elements.