But I am unable to spot any difference between that 1 good module and the rest, as I feel that the difference must be in hardware.
That is good news (that you got one to work) and bad news.
People have always wondered whether the chips on these cheap modules might somehow be fakes or rejects, as the module price on eBay is lower that the cost of the chip itself, purchased from the legitimate manufacturer.
I programmed the module for the 1Hz wave and it works just fine with Vcc applied to the board.
How about on battery power? Does this work on the alarm-output-not-functioning modules?