Why does rainy weather always appear when the cold air front comes, rather than when it is controlled by cold air?

To put it simply: rain requires air convection up and down, lifting water vapor from near the ground to high altitude, cooling and condensing, and then falling to the ground.

When the cold air front comes, it will force the warm and humid air into the sky, and the water vapor will cool and condense to form rain.

When controlled by cold air, there is no upward and downward convection power, and water vapor cannot be brought to high altitude to form rainfall.