Workflow from Illustrator to After Effects (music notation animation project)

Dear all,

I am hyper-new to After Effects but I could not refuse this job as it is for a friend so I need to learn this and just do it! I have gone through all of the ‘Learn’ in-app tutorials but, regardless, I am stuck. 

The project is about animating music notation transformation over a musical score. The process is therefore something like this:

1. prepare the notation file in the music notation program
2. export it in SVG (or PDF, but I prefer to work in SVG, unless you think PDF is better for this)
3. import SVG in Illustrator (Ai), adjust the colours and layers (every object that needs to be moved on its own layer as instructed).
4. animate in After Effects (Ae)

Now, since the Ae canvas is in 1920x1080, I’ve created an Ai document with an dartboard of the same size and imported the SVG in it. I manipulated it as needed, then saved it as an .ai file. In Ae, I’ve created a new project, then pressed Cmd-I to import, selected the .ai file and, from the dropdown, changed the type to ‘Composition - Retain layer size’. 

First problem: it is extremely pixelated. First try: activate the Continuously Rasterise option in the timeline. It improves a tiny bit but not enough. Other things I have tried from other posts here and on the wider web:

1. In Ai, turned on View > Pixel Preview and Snap to Pixel and noticed that, when enlarging, it is indeed pixelated as h***. 
2. In Ai, I doubled the size of the artboard and of its content, then saved the project and scaled it down in Ae. It is better but still pixelated and I am not sure I can work with it.

In certain tutorials on the web it is shown how a vector image remains perfectly vectorised even when enlarged, something that is not happening with my file. I believe there must be some problem somewhere either in my process or in my files. I’m attaching the .ai and .aep files for you to check whether there is something that I have missed. 

I naturally feel the deep frustration of the beginner trying to do something that is way beyond his level so, please, explain to me like I’m 5, maybe 6, no more! 

Thank you very much!