This sounds more like user error than a feature.
If you have created shapes with the parametric tools (Rectangle, Ellipse, Polygon ...) then you must convert the parametric shapes to a bezier path before you can edit the points.
To edit path points on a shape layer or mask you should have the pen tool selected(g). Trying to use the selection tool to edit vector paths is ineffective and frustrating.
Use the Ctrl/Cmnd and Alt/Option keys as modifiers.
If one point, multiple points, or no points are selected hovering over a path will give you the Add Point tool, hovering over an anchor point will give you the select point (black arrowhead) selection tool for selecting that point. Hovering over a bezier handle will give you the selection tool to grab the handle.
If all points are selected holding down the Ctrl/Cmnd tool will change the pen tool to the group select tool (white arrowhead) which will give you the option of dragging a selection around multiple points or just selecting a new single point. If you have multiple points selected pressing Ctrl/Cmnd + t will enable the transform rectangle allowing you to scale rotate and move multiple points. The escape key or enter/return key will exit the transform tool. Using the transform feature changes the tool back to the selection tool so you have to remember to activate the pen tool for further editing. I am in the habit of pressing the "g" key right after enabling transform so I don't forget to go back to the pen tool after transforming vertices. Holding down the Ctrl/Cmnd key will also change the tool to the delete point tool if you hover over a vertex.
Holding down the Alt/Option key will change the pen tool to the Convert Path tool and allow you to select bezier handles, modify the tension on Rotobezier paths or remove the curve on a selected vertex. The Add Point tool pops up when holding down the Alt/Option key when you hover over a line segment.
I don't think I missed anything. If this workflow does not work for you make sure you are running the latest build of AE. If you are still having problems let us know.