Can Lightroom synchronize the folder automatically without my having to do it manually for each jpeg edit?

I think there is a different reason: the way Lightroom is intended to be used. Being a non-destructive editor, Lightroom can export images in different sizes/quality/file formats at any time. That means there is no need to save these exported images in your catalog, because you can re-export the image if and when you need to. Export an image, use it for the intended reason and delete it after use. Export again when you need the same size/quality image again. Export with different settings if you need the same image but for a different purpose (like print instead of web).

Adobe added the option to add an exported image to the catalog because people were asking for it constantly, but it does make a lot more sense not to clutter your catalog with multiple copies of the same image and try to work with the app the way it was intended. If you need to drive a screw into a piece of wood, then you use a screwdriver. Don't use a hammer and then complain that the manufacturer apparently has no clue what screws are.