If your module has DS3231M chip then according to datasheed this modification can only generate 1Hz signal on INT/SQW pin (3) and 32768Hz on 32kHz pin(1). I have several DS3231M's and checked it.

If your module has DS3231SN chip: I had a similar issue recently. Got 10 pcs of DS3231SN from ali. Soldered one IC to my board and discovered this problem. Luckily I had ZIF SO16 to DIP adapter and built a test-board, connected it to oscilloscope and tested all of them. Only 4 pcs of 10 was able to generate different signals (1024Hz, 4096Hz etc.) on SQW pin according to datashet. Also they generated stable 32768 Hz signal on 32KHz pin (pin 1).

Other 6(also marked as DS3231SN) gave only 1 Hz signal on SQW pin, all of them produced signal on 32 kHz pin (one IC generated about 35 kHz), one IC generated floating frequency signal on SQW pin (around 1500 Hz and it cant be changed). I think it's low quality fake IC's.

This answer addresses the original question of why the interrupts didn't work. I happened to have a DS3231 lying around so I made up a test.

Interrupt vs pin number

First, this is wrong in your code:

attachInterrupt(rtcTimerIntPin, rtc_interrupt, RISING);

You need an interrupt number, not a pin number. This is correct:

  attachInterrupt (digitalPinToInterrupt (rtcTimerIntPin), rtc_interrupt, CHANGE);

I also made it a CHANGE interrupt so you get both the rising and falling pulse.

Pull-up resistor

Next, the 1 Hz output needs a pull-up resistor, so you should add that or make it INPUT_PULLUP like this:

  pinMode (rtcTimerIntPin, INPUT_PULLUP);

Testing

I used this library from Adafruit.

Adapting one of their examples, and putting in your interrupt code (with my modifications), it worked!

#include <Wire.h>
#include "RTClib.h"

RTC_DS3231 rtc;

const byte rtcTimerIntPin = 2;

volatile byte flag = false;

void rtc_interrupt ()
{
  flag = true;
}  // end of rtc_interrupt

void setup () {

  Serial.begin(115200);

  while (!Serial); // for Leonardo/Micro/Zero

  if (! rtc.begin()) {
    Serial.println("Couldn't find RTC");
    while (1);
  }

  if (rtc.lostPower()) {
    Serial.println("RTC lost power, lets set the time!");
    // following line sets the RTC to the date & time this sketch was compiled
    rtc.adjust(DateTime(F(__DATE__), F(__TIME__)));
  }

  // enable the 1 Hz output
  rtc.writeSqwPinMode (DS3231_SquareWave1Hz);

  // set up to handle interrupt from 1 Hz pin
  pinMode (LED_BUILTIN, OUTPUT);
  pinMode (rtcTimerIntPin, INPUT_PULLUP);
  attachInterrupt (digitalPinToInterrupt (rtcTimerIntPin), rtc_interrupt, CHANGE);
}

void loop () {
  if (flag) {
    digitalWrite(LED_BUILTIN, HIGH);    // flash the led
    delay(100);                         // wait a little bit
    digitalWrite(LED_BUILTIN, LOW);     // turn off led
    flag =  false;                      // clear the flag until timer sets it again
  }
}

Question

How to use a 3V logic signal to switch on/off a P-Channel power MOSFET which in turn switches on/off a 5V power supply for Arduino, Rpi, or ESP32?

Answer

(1) I must first point that you don't use DS3231's Square Wave Signal to switch/trigger the power MOSFET or other devices. You use the interrupt signal.

(2) To switch/trigger a power MOSFET switch, either P-Channel or N-Channel can do the job, as shown in the following schematic.

Discussion, Conclusion, and Recommendation

(1) Using DS3231 to control the power MOSFET which in turn to switch on/off might not be a very solution. My usual trick is to use DS3231 interrupt to direct switch on/off the LM2596 PSU through its power on/off switch pin. See Appendix B for more details.

(2) Using P-Channel or N-Channel power MOSFET, High or Low side switching?

I usually prefer NPN BJT and N-Channel power MOSFET for switch, because (a) more EE guys prefer NPN and N-Channel so you will find more tutorials to follow, and circuits to copy, (b) N guys are more efficient, (c) In case you need to ground complicated circuits, N-Channel and NPN BJT low side switch can ground easier. On the other hand, high side mosfet switches "floats" in the air, so not so easy to make a common ground.

(2) The Sony 18650 tech spec is the best I can google.

(3) The OP's step up regulator is interesting. I bought one long time ago but never tried it, because my projects don't have any space problem. I usually use 18650 x3 or even x4 and step down to 5V. The switching power supply regulator has a very high, or 90% efficiency, so your PSU does not loses that much energy, but lasts 3 or 4 times longer.

(4) I think the OP's design is very good. My suggestions are a bit subjective. The OP need to do engineer trade offs and cost benefit/risk analysis. It is a good learning experience and fun trying different approaches, compare and contrast.

References

(1) IRF9540, SiHF9540 P-Channel Power MOSFET (Vgs(th) = -2V min, -4V max, Rs(on) = 200mΩ @Vgs -10V) datasheet - Vishay

(2) DMP2008UFG P-Channel Power MOSFET (Vgs(th) = -0.4V min, -1.0V max, Rs(on) = 17mΩ @-8.3A) datasheet Diodes

(3) Sony 18650 Lipo Battery Spec

(4) Amazon USB DC-DC Converter 0.9V - 5V Boost to 5V 600MA Step Up Power Supply Module

(5) DS3231M ±5ppm, I2C Real-Time Clock Datasheet - Maxim

Appendices

Appendix A - DS3231 Testing Notes

I am reading my old post to show (1) how to use Rpi OS's DS3231 DT drivers and also (2) basic python program for setting up and also how the program the interrupt for alarming external devices as asked by the OP. I have also used a scope to display the DS3231 1Hz and 32kHz wave forms and found them a bit noisy.

Rpi3B DS3231 pythonProgramming Notes

(1) https://www.raspberrypi.org/forums/viewtopic.php?f=37&t=77158&p=1350076&hilit=DS3231+tlfong01#p1349687

(2) https://www.raspberrypi.org/forums/viewtopic.php?f=37&t=77158&p=1349937&hilit=DS3231+tlfong01#p1350076

(3) https://www.raspberrypi.org/forums/viewtopic.php?f=37&t=77158&p=1349937&hilit=DS3231+tlfong01#p1349687

Appendix B - Using DS3231 interrupt signal to switch on LM2596 PSU for Rpi

Using High/Low side power MOSFETs to switch on off PSU might not be a very good solution. One trick I always use is to use DS3231 interrupt signal to switch on/off the LM2596 PSU, or using the interrupt to switch on LM2941 current switches.

Rpi freezes every now and then, how to fix it with a watchdog?

I modify ordinary DC-DC (12V to 5V) PSUs' so that any Rpi or MCP23x17 GPIO pins can power on/off the LM2956/LM2947 voltage regulator chip of the PSU. (LM2941 can be used for 1A current switches, LM2596 for 5V 3A PSU. The on/off pin is also connected to a push button, for manual power on/off testing.)

Actually each of my 7 Rpi3B+'s is connected to a cheapy DS3231 Real Time Clock Module which has a hardware interrupt pin to reset PSU, Rpi, or other devices.

Now the external DS3231 RTC wakes up everybody in the morning, and switches off lights at midnight, so everybody goes to bed.

End of Answer