The reason why the air wouldn't flow from one side of a spinning sphere to the other is due to the phenomenon known as the Coriolis effect. The Coriolis effect is an apparent deflection of the path of a moving object caused by the rotation of the Earth.
When you're standing on a spinning sphere, like the Earth, the air around you is also rotating along with the planet. As a result, the air near the surface is influenced by the Coriolis effect. The Coriolis effect causes moving air (or any moving object) to appear to be deflected sideways from its intended path.
In the case of the Earth, the rotation causes the air to deflect to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. This deflection creates global wind patterns, such as the trade winds and prevailing westerlies, which are responsible for the circulation of air around the planet.
So, when you're standing on a spinning sphere like the Earth, the air around you is also moving with the same rotational speed. As a result, there is no net flow of air from one side of the sphere to the other because the Coriolis effect deflects the air sideways, causing it to circulate around the planet rather than flowing directly across it.