> Your linked video maps the 4th dimension to time, it doesn't project to 3D
No, it projects it to 3d. The movements you see are rotations, not movements of an intersection plane. You can clearly see at any point in time each of the 8 identical cubes making up the tesseract, skewed and resized by projection to a 3d perspective.
Your linked video maps the 4th dimension to time, it doesn't project to 3D. Projecting through time (especially non-interactively) lacks the immediate feedback needed for the brain to grasp it as intuition.