Think of the Gradient Stroke in AE as a gradient on a comp sized solid with a stroke as a mask. It's completely different than Illustrator's gradient stroke. The path in Illustrator becomes the path for the gradient.
If you're using just a single instance of your Illustrator path in AE you can just bring in Illustrator artwork. If you want to use shape animators there may be a workaround. You can tie the position scale and rotation property of duplicated layers to the controls in the shape layer repeater function. You just have to duplicate the layer as many times as needed. The expression is a little tricky for me to create outside AE from my head.
If you want to use trim paths then back in AI, duplicate your gradient stroked illustrator layer, put the duplicate on a separate layer and change the stroke to a solid color. Open the .ai file in AE as a comp, convert the duplicate stroke layer to a shape, apply trim paths to the shape, then use the shape as a track matte for your gradient stroke.
If you need to use one of the other shape layer animators let us know. There may be a workaround.