Shrinkwrap modifier works well within rigs, at least up to 2.79. Many if not most of my character rigs make use of it at some point.
If the piece of cloth is left behind you probably did not rig it such that it follows the target during animation? Because this is what you need to do in order to have it work properly: the cloth needs to have an armature modifier too, and must be skinned to follow suitable bones during animation. The shrinkwrap modifier has to be placed after the armature modifier in order to wrap the already deformed mesh around the deformed target. How exactly the modifier will behave is a matter of the mode you select as well. If the cloth is left behind you probably used ‘project’ mode, ‘nearest vertex’ or ‘nearest surface’ will always pull it towards the target mesh, however the mesh needs to be in a suitable start position before the pull kicks in to have it wrapped around the target in a cloth-like way, so the above is valid here too.
The shrinkwrap constraint btw. will pin your cloth object to the target’s surface at object level, ie. without deforming the mesh. So you were right, the modifier is your friend.
Apart from that, I’m not sure whether it is a good idea to learn new rigging tricks using 2.8, there’s just too much stuff not yet in place. I’d use 2.7x and get it up and running there first, the errors you encounter will probably be your’s. With 2.8 on the other hand you’ll never know …