Importing Path from illustrator to After Effect

When you need a stroke in Illustrator to become an open path in After Effects, you must use a simple stroke in AE. No Brushes, No gradients. Nothing but a stroke. In your example, you need to create a new layer, copy the yellow X, and paste it into the new layer. Name the layer Track Matte, remove the brush from the layer and ensure it only contains two simple paths. The stroke only needs to be a couple of points wide, but you should extend the path beyond the original X on both ends.

You now have three layers in your AI file. The phone layer is on the bottom. Above that is the original yellow X, then the Track Matte layer with a simple stroke over each X.

Import the AI file into After Effects as a composition retaining layer size. Open the comp created with the import, convert only the Track Matte layer to a Vector layer, press the U key twice to reveal all modified properties of the layer, and ensure that you only have a simple path for each part of the X. Now you can change the end cap for the paths and the stroke width so that the lines are just a little wider than the original "X," add Trim Paths, and set the Track Matte layer you created as a Track Matte for the original X layer. 

For an illustration that is this simple It would be faster to just import the AI file as a composition so that the Phone and the X are on separate layers, then use the Pen tool to draw a simple shape layer above the X layer and use that as a track matte.