So you have a 16A fuse/breaker, and a wire that will take 16A. Then you plug in your lamp with a thin wire that can only take 3A. In the UK that's fine, as you have a 3A fuse, the lamp develops a fault and starts pulling 10A, the fuse blows, job done.

Without that fuse though, the wire overheats, and you get a fire.

Looking at the fairly new soldering iron plug next to me, with a 3A fuse that wire does not look anywhere near thick enough to qualify for passing 13A.