Pixelated vector even after I toggled continuously rasterize

It's basically everything Rick said - your lines are to thin and the pattern being aligned perpendicularly doesn't help. At the very least use an angled pattern fill. That way it will get antialaised all the time which would at least mitigate the flickery Moiré patterns somewhat, but otherwise you literally need to get back to the drawing board and rework your design.

Mylenium

I see a few newbie problems. They start with the design.

Single-pixel (1 pt) lines and fine repeating patterns never work well in video. Thin horizontal and vertical lines must be lined up with the pixel grid in AI. You can turn"snap to pixel" on in the settings. You should always run a pixel preview of your artwork in AI and check it at 200% to 400% to make sure the lines do not fall apart. 

All elements that you plan to animate should be on a separate layer and they should be sized and positioned in the hero position. The hero position is the position of the elements in your design in their resting place. If you want the turkey to be on the conveyor belt then place it there in the middle of the belt in AI. 

When you import this kind of artwork you should import it as a composition retaining layer size, then, and this is IMPORTANT, open the comp that After Effects creates. Your Illustration will be in the hero position with all layers properly positioned and ready to animate.

There are several reasons that you need to create the artwork with Snap to Pixel turned on, keep thin lines at least 2 pixels wide, and do not use small repeating patterns with thin lines, and put your illustrations in their hero position. Anything that is not aligned with the pixel grid will be antialiased and the edges will soften. You can get away with that when the lines are moving, especially if you enable motion blur, but when they stop moving thin lines must line up with the grid. You can't use 1 point lines because they fall apart when the video is compressed and most folks will be looking at your video in a media player that scales it up or down to fit the device, screen or web page. 

You can't use thin line repeating patterns because they look horrible when they move, are subject to moray patterns, don't compress well to delivery formats line H.264, and they just don't look good when scaled.

The size of your AI Artboard should match the comp size. You can scale up or down without losing quality but if you have an element that has a 2 point stroke you should never scale that element (layer) down below 100% unless you can live with the lines falling apart. 

None of the line size or pattern size problems you will run into have anything at all to do with CR being turned on. The vectors are going to be turned into pixels and if they do not line up with the grid perfectly, they will change color. 

Put a 1 pixel wide white vertical line in a comp, Magnify the comp panel to 400% or more, and check the edges as you try and get it lined up with the pixel grid,  As you move the line around it will change from 1 pixel wide solid white to 2 pixels wide with one column a little gray and the other darker gray. When the pixel grid has been split you'll have a 2-pixel wide gray line. As it continues to move the pattern will repeat until you have a 1-pixel wide white line again. That is just how video works. Here are 5 small rectangles with a 1-pixel white stroke. You'll see exactly what I mean:

Run that image through the Media Encoder and render an H.264 file for YouTube and the lines will even fall apart more. 

I hope this helps. There are a few old articles on preparing artwork for video. These are just some of the points they cover. unfortunately, I don't have a link I can share right now. Maybe somebody elce will share one.