Single Path: Image Tracing

You can use Animate CC to make it one path. Import your Illustrator file into Animate with Place objects at original position selected. Use Modify > Break Apart (Ctrl + B) to convert the drawing objects into paths. With snapping turned on, drag an endpoint of a segment to snap to the path where you want it to connect. You're still manually connected these, but I find it faster in Animate with snapping on and you can connect multiple segments. When it's connected, it will be one object. You'll know it's all connected if when you click on a portion of the path, all of it shows that it's selected. Save it out as an SVG. Then import the SVG into Illustrator to save it out as an .ai file so that you can import it into After Effects.

Claire,

You may wish to simplify the structure of the traced paths by removing bits, such as one in the double curved part to the left.

As you have found, you can only have two segments meeting in each Anchor Point.

The only way to have a single open path running along all the present paths is to run over some of them twice forwards and then backwards (you can create a copy and then join them at the farthest/end Anchor Point) to get back to the starting Anchor Point (creating what corresponds to an invisible loop), thereby ending with another coinciding one on top that you can go on from.

Whoo boy! This one's challenging. Image trace isn't perfect, nor is it really made for what you're wanting - but you've come close. The way to do the blending of multiple paths into on, WOULD be the join option, but as you mention, that's not effective for this purpose. In fact, I kind of think that any automation further than where you've gone will take longer than what I'd suggest. I'd be happy if others have a better idea. Here goes:

With the Trace completed, go to the Layers panel and lock that layer - you may even want to double click the layer and tell it to be a "tracing layer" in the resulting dialog box (that will make it more translucent, if that helps you for the next step). Then, on a layer above your trace, trace this AS a single path. Use the pen tool, if you're proficient with it, or even use the pencil tool (which will create WAY too many vector points, but will allow you to draw "by hand"). The resulting layer will contain exactly what you need for your AE trim paths, and although it's cumbersome, it's likely the quickest available way to end up with the result you want.

Otherwise, I know of no other commands or plugins that'll do specifically what you need.