No. Non-polarized light is a mixture of light waves which oscillate in all possible planes. When it enters a scattering medium these waves will be subject to being scattered. Some light waves will travel further through the medium before they are scattered. in the medium is not a strong scatterer, and the total distance traveled through the medium is short enough, much of the light will make it through without being scattered. If it is a strong scatterer, or the light must travel an extended distance through it, less of the light will make it straight through and more scattered along the way.
Any light wave scattered will scatter in a direction that is on a plane that is at a right angle to the plane that the electric field is oscillating in. Since non-polarized light contains waves that are oscillating in all possible planes, you will end up with light waves being scattered in all directions.
When you polarize light by passing it through a filter, you are only letting light waves through that oscillate in a limited direction. When this light enters a scattering medium, just like above, it is subject to being scattered. And like above, some waves will travel further before being scattered. But since the direction in which the light waves are oscillating is restricted, so is the directions in which the light waves will be scattered. This is what the video is demonstrating.