If I only use the Arduino Nano with the DS3231 and run the I2C Scanner of Nick Gammon,
it detects the RTC and the eeprom of the RTC:
If you use that variant do you power the DS3231 by 3V3 too? The DS3231 is 5V safe, so you can power it by 5V and if you do that you don't over-voltage the I2C signals.
The Nano's 3V3 pin is powered by the FTDI chip so you must not draw much power from it.