Hello again,
I programmed the module for the 1Hz wave and it works just fine with Vcc applied to the board.
Thankfully, the connection from the IC SQW pin on the DS3231 was intact when tested with a multimeter.
I don't have an unmodified version. All of them have resistor pack 1 removed.
Although, it is worthy to state that, with your previous suggestion of an external pull-up to Vcc, one out of 5 of my ZS-042 modules generated interrupts on battery power, as desired! Also, all of them were running on the same code.
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.
@Edit:
On further testing with the non-functioning modules, I saw that the flags for the interrupts get set (A1F and A2F correspondingly) in the status register but the INT pin does not go low hence the arduino does not see an interrupt.