Is atmospheric pressure balanced on an object as it moves?

If a body is moving very slowly, the air can easily move around it, resulting in the negligible drag terms. In practice, that void gets filled by air near the void, and the atmosphere takes the least effort required to populate the empty space. When you move the box, the air in front of the box is displaced, creating a slightly higher pressure region, which induces a flow towards the back of the box with its slightly lower pressure. We're talking tiny fractions of a psi, not the full 14.7psi of atmospheric pressure. Go slow enough, and the pressures will be extremely low. You will get the result you found: we can neglect drag and just say work-in = work-out

Increase the speed, and we know this process becomes non-negligible. It's a major design factor for cars and airplanes.

If you want to feel what it would be like if the air did not flow like this, find a syringe without a needle. You can move the plunger in and out without too much difficulty. When you pull the plunger out, the air above the plunger is at a slightly higher pressure, the air inside the syringe is at slightly lower pressure, and a flow is induced from above the plunger to into the syringe. When you push the plunger, the opposite happens.

Now completely push the plunger in, and cover the tip with your finger. Then pull on the plunger. Now there's no path for the air to move into the syringe, and you can feel what 14.7psi feels like!