I think there is some better explanation from stand of point theoretical fluid dynamics.
Yes, there is, but whirlpools are mostly irrelevant to the shape of Earth.
But the wind itself also follow rotation of the earth because of conservation of angular momentum. The wind carry tangential velocity from rotation of the earth before it become a wind. If not then beside it travel to north or south it will have additional motion to the west up to 465 m/s.
You are somewhat right.
The air does start by moving with Earth. So say you have air at the equator, travelling at 465 m/s, to the east, making it move along with Earth.
But now air pressure differences causes it to become wind moving to the north (the exact speed doesn't matter that much).
Now, with it moving to the north, it is still travelling some 465 m/s east. But now the ground it is above is only travelling at 460 m/s.
This is going to make it appear to drift east or curve to the right as the east-west component of the wind is now greater than the east-west component of Earth's motion.
As it continues moving up, it will continue to drift to the east and curve around until its directionality is pure east.
What if the air was moving south towards the equator from the north?
Well lets say you have the air start off moving at 400 m/s east, along with Earth. But now it moves south, to a region where Earth is spinning at 405 m/s. This means the air is now travelling 5 m/s slower than Earth and thus it will fall behind, drifting to the west, or apparently curving to the right (as west is right of south).
These apply regardless of if it is moving perfectly north-south or off that with an east-west component as well. In the northern hemisphere moving north or south will cause the wind to appear to curve to the right.
But what about when the wind is moving due east or due west? Well this is somewhat more complex.
First you need to note that it will be following a great circle route for a "straight" path, not continually moving east/west. In a large scale example this will result in your path turning towards south just to follow the straight line. But in the same time you have moved this distance, Earth will have rotated to the east and thus you end up west of your apparent straight line path. This results in you curving right.
So regardless of how the wind moves, assuming it starts off pretty much following Earth, it will curve to the right in the northern hemisphere.
In the southern hemisphere it will curve left.