Thanks! Although not yet 100% stable because I improvised without soldering the direct connection E- to GND, the E- now 0V and A+ and A- are 2.5V, but the scale definitely change values once I press a load. So I would say indeed root cause is HW issue with the HX711.
Regarding "They don't have a double-sided board, so the bottom copper ground plane is missing. You can see that by holding the board against a strong light (faulty ones are transparent)": From the picture I attached above, is it indeed considered transparent?