I think the point is that the service mesh can't do the work of propagation. It needs the client to grab the input header, and attach it to any outbound requests. From the perspective of the service mesh, the service is handling X requests, and Y requests are being sent outbound. It doesn't know how each outbound request maps to an input.
So now all of the sudden we do need a client library for each service in order to make sure the header is being propagated correctly.