Not sure what else to try. Clearly the microcontroller and Ethernet controller are both running fine, so VCC is clearly ok. The infinite resistance should be ok (the reading may vary depending on the multimeter, as long as the voltage is about 3.3V it’s fine). The LCD you received has been tested and verified to work.
The LCD backlight and contrast are each controlled by a microcontroller PWM pin. The fact that you see backlight on means at least one of the PWM pins is sending signal correctly to the LCD (otherwise the LCD will remain dark). So this leaves some of the other LCD pins to check. One thing I suggest you do is to measure the voltage on each of the LCD pins: if you look at the LCD, there are 16 pins, labeled VSS, VDD, VO etc. VSS is ground. Could you measure the voltage from each of the pins 2 to 15 to ground, and write down the voltage value? This way I can compare to the reference voltage and tell which pin is having a problem.