Vertices of imported Illustrator vector graphics cannot be animated

I'm amazed. A tutorial with 150K + views only demonstrates how to convert a layered AI file that you failed to import as a composition into a layered composition, which works but does not offer the options of choosing document size instead of Layer size as footage dimensions. While it is true that layer size is often the most useful option, document size makes some designs easier to handle. There is no explanation of why you would want to convert AI layers to shape layers, the reasons or proper techniques for preparing layered AI files for animation or what you can do with the AI file once you import it. I would give the tutorial a D. 

Let's get to your problem. This is the screenshot you posted:

From your question, I think you want to use Puppet Pin to animate the hair while leaving the rest of the face alone. Puppet Pin has nothing to do with Anchor Points, but it has everything to do with layers. The preparation of the artwork in AI is the critical part. Each element you want to animate must be on a separate layer. The layers cannot be nested inside another layer. If you properly set up your layers in AI with the bottom layer being the background, the next layer being the pinkish shapes that appear to be behind the hair, then layers containing different parts of the hair so you can animate the different strands individually, then the face, then the eyes or other facial details you may want to animate, then the hair that is going to move in front of the face, then any layers that appear on top of the hair using blend modes or transparency. Once you have the layers all separated out you import as a composition and in this case retaining layer size is the right choice. Forget just importing the file, creating a comp from the merged layers, then creating a layered comp, then opening that and converting all of the layers to shape layers. None of that is necessary to use puppet pin to animate the hair. 

Once the composition is created you pick the layers that you want to animate and apply puppet pin to those layers and create the movement. You do that one layer at a time with every layer you want to animate. A little fine-tuning with using blend modes and maybe some slight color changes and you are done. If there are a few strands of hair on a layer that you might want to animate the path of the layer you can convert just those layers to shape layers, then press the "u" key twice to reveal the paths on the shape layer, then click the stopwatch on the paths that you want to animate and carefully set a few keyframes to change the actual shape of the paths. Beware that animating vector paths is a tricky business and requires a fair amount of practice.

When preparing your AI file for animation it is a very good idea to name your layers and give them names that really mean something. It is also a very good idea to set the artboard to the same dimensions in points or pixels as your intended delivery format. (HD 1920 X 1080 for example) It is also extremely important that none of your artwork extends beyond the artboard and that you do not have multiple artboards in the AI file. Another good idea is to turn on Pixel preview and Shap to Pixel in AI so that your artwork ends up lining up on the pixel grid. 

I hope this helps a bit. Make sure you vet your trainers. Most tutorials out there are prepared by well-meaning enthusiasts that don't really understand efficient workflows, leave out critical information and often give really bad advice that does not help you in the long run. Become familiar with the User Guide and when you want to start using a specific tool or effect that you are not familiar with it is almost always a good idea to type the name of the tool or effect in the Search Help field at the top right corner of AE. This will open up the User Guide, which is pretty good, and also point you to forum discussions and other resources on things like Puppet Pin.

By the way, you can use the Anchor Point, sometimes called the Pan Behind tool, to reposition an anchor point at any time without changing the position of the layer by simply pressing 'y' to select the tool then selecting and dragging the Anchor Point in the comp panel. If you move the Anchor Point by changing the values in the timeline the layer will change position.

I hope this helps. Here is the link typing Puppet Pin in the Search Help field will bring up: Puppet Pin You will find a few much better tutorials on the workflow right at the top of the page.