I think the potentiometer works because it gives diffrent voltage to the LCD. It also changes the contrast a liiiiiitle liiiitle bit.
You should be able to vary the voltage at pin 3 from 0 to 5 Volts (with respect to pin 1). The display should be blank at +5 Volts and you should have a single solid row of blocks at 0 Volts. The proper setting, which will still show the blocks, will be around 0.4 Volts for your LCD.
Don