Have a shape follow a path (which is animated with trim path)

The Trace Path option in the bundled script, Create Nulls from Paths will do this easily. The script can be found in  a subfolder where you installed AE. The subfolder will be here - Support Files>Scripts>ScriptUI Panels.

Here's a quick tutorial to show how to invoke the script and to set up the animation.

https://youtu.be/Rp40MieeFDM

If you ease the in and out points for trim paths the easing will exactly match easing the position keyframes of the layer following the original path. I do this all the time. 

I checked your comp and it has a few basic problems. The most obvious problems are with the Circle layer and the Path layer. If you want a motion path to follow a path the transform properties of the layer with the path must all be reset. Your Path layer is all fouled up. You need to reset Position and Anchor Point so that the Anchor Point is at 0, 0 (true for all shape layers), and the position is 960, 540.  Now when you copy the path in the path layer and paste it to a motion path it will line up. 

The second problem is with the Circle layer. The Transform Ellopse Path 1/Transform property must be reset, You also need to reset the Anchor Point and Position property of the Circle layer.  This will put the circle in the center of the comp. 

The next step is to set a keyframe for Ellipse 1/Path 1 and copy or cut it to assure that you have copied the path to the clipboard. When that is done, set a keyframe for the Circle/Transform/Position property at the first frame, the same frame that the first keyframe of Trim Paths 1/End is on, and then paste. This will give you Rove Over Time keyframes for the Circle/Transform/Position property. Press the K key a few times until you get to the last keyframe for Path/Contents/Ellipse 1/Trim Paths 1/End and then drag the last Position keyframe for the Circle layer to the same frame. It should snap into position.

Because you used End instead of Start for Trim Paths you will need to select all of the Circle Position keyframes and use the Keyframe Assistant to Time Reverse the keyframes. There is no ease and whiz expression applied to Trim Paths 1/End, but there is easing so you should select that keyframe again, press f9 to set easy ease, or use the keyframe assistant to set easing for that keyframe. The last step is to select the last Position keyframe for the circle layer and press f9 or use the Keyframe Assistant to set the same easing.  If you need to further refine the easing you can right-click on any keyframe and select Keyframe Velocity to manually set the easing. You just have to copy the keyframes. Ease and Whiz will not work when there are Rove Over Time keyframes on a property so you'll have to adjust that easing manually.

The last step is to adjust the camera animation to properly frame up the shot.  

One more point, Your comp was set to have a white background. In almost every case, this comp will render with a black background. If you want a colored background you have to add some pixels on the bottom layer. The easiest solution is to just add a comp sized solid.

That will fix your problem. Just remember that any time you want to have a motion path follow a vector path the layer with the path must not have any of the transform properties modified. 

Here's the fixed project file. Took me longer to write the post than it did to fix the problem.