Looking at your 4th picture, it seems that there is a metal tube running up the inside. I think the argument is that it doesn't need to be earthed because the cable that runs inside it is double-insulated, i.e. it is standard flex where the individual wires are insulated and then there is an outer sheath over everything.
In contrast, sometimes you get light fittings (e.g. chandaliers) where the wires inside the metal tubes are singles, often with heat-resistant insulation; in these cases, it's necessary to ensure the metal is earthed.
If you decide to replace the wire, use regular flex, not single-insulated "bell wire" or "speaker wire" like you currently have connecting to the plug, and check for any sharp edges on the metalwork while you're re-assembling it.
Personally I'm always a bit cautious about lamps like this because of a family story about someone who was killed by one. I believe that a cleaner was in the habit of always tightening it up, and in the process the wire became twisted and eventually failed; the victim was killed while turning it on. This was long before the invention of RCDs, of course.