Best to replace the cord with a grounded one, and use a cable gland where the cord goes in to act as strain relief. You do get inline light switches that will accommodate small diameter 3 core cables, and there you only switch the active ( brown) conductor, the others passing through in a internal slot.
Grounding the fixture is good from a safety point, and also has an improvement in enabling the lamp to start easier with the capacitive coupling along the tube length, though those small 8W tubes do have a short life, simply because they have a very low mercury dose, and tend to have a lot of power cycles in a bedside lamp role, which strips the cathodes rapidly.
however that tube in there is easily 30 years old, and probably is either Sylvania or GE made, but those plants closed 20 years ago.