EEPROM.put() endianness

Are you sure about that? Consider a Teensy 3.2 (based on ARM Cortex-M4 ):

void setup() {
  Serial.begin(115200);
  delay(1000);
  
  uint32_t x = 0x12345678;
  uint8_t *ptr = (uint8_t *) &x;
  for (uint8_t i = 0; i < 4; i++) {
    Serial.print(*(ptr + i), HEX);
    Serial.print(" ");
  }
  Serial.println();
}

void loop() {
}

Output:

78 56 34 12 

Same code run on an Uno:

78 56 34 12