Hi wily,
You are quite correct; drawing unbalanced current from the mains supply is not allowed, in the UK anyway. But have no fear the power that a single string of LEDS takes would not cause a problem. If there are hundreds of thousands, then there would also be hundreds of thousands of mains grid circuits, so once again no problem.
Untill relatively recently many domestic electrical items were half-wave rectified: mains radios (50W), CRT TVs (200W) and so on.