If you can post a couple pictures of the carburetor and choke plate, it might help. If you can move the choke plate by hand I'm guessing the choke cable (stiff wire) may not be connected to the choke plate OR the choke sheath is not bolted to the carburetor or choke lever bracket. If you know this already, please disregard - the choke cable is basically a stiff wire inside a tube (sometimes its called a sheath or a jacket but basically it is a flexible hollow tube). The tube needs to be firmly fastened at both ends for the stiff wire inside to move the choke, The stiff wire should be connected to the choke lever at the dashboard and the other end of the stiff wire connects to the choke plate. When the choke control is pushed in or pulled out, the choke plate moves correspondingly if the sheath is firmly fastened. It is very rare that the stiff wire breaks so check to see if the stiff wire is connected at both ends.
If the wire is connected at both ends, watch the sheath as you move the lever. If it bounces or slides, most likely it is not clamped at the choke control at the dash or at the carburetor. Post some pictures if you cannot figure it out.
If the wire is connected at both ends, watch the sheath as you move the lever. If it bounces or slides, most likely it is not clamped at the choke control at the dash or at the carburetor. Post some pictures if you cannot figure it out.