Thanks! Wow, this is a tough one for sure. I assume that the 77º orientation refers to the larger rectangle on the left. Even you said that's the orientation of the tail, I wouldn't think it is as simple and reliable as to take the opposite of it, i.e. 77º + 180º = 257º, as the orientation of the head.
I am not aware of a tool in ArcGIS Pro that recognizes a pointy shape. Simplify Polygon can reduce redundant vertices, leaving major/sharp turns in the shape. But then you still need to calculate turning angles at each vertex and find the sharpest turn with two long joining segments to be a pointy end.
For the shape on the left, you may hope the area in the green circle is the pointy end, but it can be easily similar or confusing with the areas in red circles to an algorithm, I would say.