How to parent a point of a shape to a layer

Yes. This for the win!

1. Window -> Create Null From Paths

2. Tunnel down in your shape layer and select the most deeply nested "Path" property, which has the stopwatch icon.

3.) In the "Create Null From Paths" window, click "Points Follow Nulls".

You should now have nulls for each point of the path, and you can individually animate those nulls. You can parent those nulls to anything you'd like.

The last time I used a shape, I noticed you could see the path vertices and the anchor point when highlighted.  It seems like a simple chore to move the anchor point until it's over one of these vertices.  It should then be a simple matter to make a null, and using the shape layer's anchor position property, copy-paste the value to the null's position.  Parent the shape to the null and you're good to go.

I estimate the time it would take to do this at about two minutes, including mistakes.

Would that work well enough?

There is a $35 plugin called Motion by Mt MoGraph that has a function called "rope" that will create a highly controllable line between any two objects.  I believe it uses the beam effect.  So you could create lines between a series of nulls then parent the nulls to each other in a skeletal hierarchy and you've got yourself a rig for a stick figure. The script is well worth the money as it does many other incredibly useful things. It's like a swiss army knife for After Effects .  Just do a Youtube search for Motion 2 for After Effects.  There's a demo of all it does.  It's the best $35 I've ever spent.

You could also just parent shapes to nulls & then nulls to each other to create a simple rig but it will only control rotation and nothing else.  If you want

squash and stretch you'll have to start digging into beam effect or just invest the $35 in Motion.