Center Anchor Point for Contents within Shape Layer

This seems like an awefully simple thing to want to do, but for some reason is impossible without needing lengthy expressions. 

This script will do what you want: https://blob.pureandapplied.com.au/a-content-centric-model-of-the-universe/ (put it on KBAR)

Putting an expression on is not really time efficient and complicates things if you are doing it for work e.g. creating a simple 2 line crosshair on one layer would take you ages.

Parametric shapes are easy. Press the U key twice and reset the Contents/Rectangle/Transform Rectangle property to 0,0. 

Paths are a different problem. If you reset the anchor point, the layer will move because the position will not automatically change like it does when using the "y" Pan Behind tool. You can use an expression to center the anchor point on any shape layer by using sourceRectAtTime(). Left will give you the left edge, and the width will provide you with the width, so half the width + the left will give you the center. Top will give you the distance from the layer's anchor point, and height will give you the height. 

Apply this expression to shape layers Anchor Point:

ref = sourceRectAtTime();
w = ref.width/2;
h = ref.height/2;
l = ref.left;
t = ref.top;

[w + l, h + t]

That will center any path or parametric shape on the Shape layer's Anchor Point. If you have more than one shape on the layer, it will center the entire group. Only the Anchor Point will change, and the shape layer contents will be repositioned. 

Unfortunately, sourceRectAtTime() does not take stroke width into account. As long as you have a closed path,  the anchor point will be in the center of the shape, but if you have a C-shaped or U-shaped path, the anchor point will be off by half the stroke width. 

You can use a similar approach with slightly different modifications to the expression to position the Shape Transform anchor point of a shape in the Contents section of a Shape layer so you can scale, rotate or skew from any corner or the middle of any side. I use this expression all the time. This is the one I have stored as an animation preset. It requires adding a Dropdown Menu Control and naming it Anchor Point Control, and adding nine menu items. It will work on any layer.

ref = sourceRectAtTime();
w = ref.width;
h = ref.height;
l = ref.left;
t = ref.top;
ctrl = effect("Anchor Point Control")("Menu");

if (ctrl == 1)
	[l + w / 2, t + h / 2];
else if (ctrl == 2)
	[l, t];
else if (ctrl == 3)
	[l + w / 2, t];
else if (ctrl == 4)
	[l + w, t];
else if (ctrl == 5)
	[l + w, t + h / 2];
else if (ctrl == 6)
	[l + w, t + h];
else if (ctrl == 7)
	[l + w / 2, t + h];
else if (ctrl == 8)
	[l, t + h];
else if (ctrl == 9)
	[l, t + h / 2]

I use it often. It is probably one of my top 10 Animation Presets.

You can do it manually by using the Pan Behind Tool.

With the shape selected activate the Pan Behind Tool (Y) and you can drag the anchor point of the shape wherever you like.

You could also open the Transform properties of the shape (it will say Transform: Shape name) and reset Anchor Point to 0, 0. But then you will proably have to adjust the shapes position when it shifts.