There is usually a way to do what you want in AE if you think outside the box. If that happens you can add a stroke, then just turn the stroke off in the AE project. Sometimes AE is buggy creating shape layers from vector layers. Here's a zip file for you with both a CC and a CS6 project inside. If you use Trim Paths then you must have a straight tapered line and you have to set the right first vertex and then animate both start and end values to make the line start at the thinnest end and extend to the thickest end. If you use Offset Paths instead of Trim Paths then the line starts growing from the thinnest point and moves to the thinnest point. After Effects series BEGINNER 8 MIN Create a handwritten title Animate lettering in Adobe After Effects so that it appears to write by itself. Adobe After Effects has a number of tools that allow you to quickly add and modify a text stroke. When it comes to working with text, the stroke can be a powerful design element. There are two ways to animate the creation of a tapered line. In this video tutorial, learn how to create a customized animated stroke effect in After Effects for your next video production project.
The way to do this is to start in Illustrator, create your tapered lines there, Expand appearance in Illustrator, then import as a comp in AE and convert vector layers to shapes.