Yes, The fuse in a mains plug is there to protect the cable from the plug to the device from overheating or melting, so the fuse needs to be rated for the cable size, not the plug capacity. Not many plug connected household appliances draw 13A, which is about 3 kW ! But too many small devices are, or used to be, supplied with a 13A fuse.
My understanding is that unless a device is double insulated, (and shows the double square logo), any external metalwork should, ideally, be connected to earth.
The earth lead is there to carry sufficient current to blow the fuse in the plug if the metalwork becomes live, thus disconnecting the device from the supply.
Edited By John Doe 2 on 20/04/2022 12:07:12