It seems like you suffer from a misconception. "The left eye sees more of the left side of an object..." is not how distance perception works. Otherwise we wouldn't be able to estimate the distance from flat objects, such as traffic signs and shooting targets.
The actual mechanism is parallax estimation, or Binocular disparity. In a nutshell, the closer an object is to your eyes, the bigger the difference of its position on the left and right eye's retina will be.
You may perform a simple experiment: find a place where several parallel wires hang in the air: a train line or aerial phone/power line. Look at the wires normally, and they will appear just as black lines in the sky, with no distance perception. Now tilt your head to the side, and you will instantly gain a feeling which one is closer, and which one is further away. Why the difference? Because any shift of a horizontal wire gives the same picture, meanwhile for vertical wires the difference is obvious.
When you close one eye, your brain loses the ability to estimate parallax. However, it is left with several options:
- Visual cues. If one object overlaps another, it's obviously closer. If two objects look the same but one is smaller, it's probably further away (or a child). If you know the expected size of an object (a mountain/a horse/a fly), you may get the feeling of distance as well.
- Focusing distance. When you focus your eye on a near and far object, the feelings are different.
- Memories. You remember what you've seen with two eyes.
Of these, only (3) depends on previous binocular vision. (1) and (2) are available even to those who were born with one eye. However, parallax estimation is much more fast and precise. With a single eye you will be able to hit a fly on the wall, but catching it in the air will be extremely difficult.