Ok… I guess, strictly speaking, I did only ask about output.
However I think things may go faster with a bit more verbosity on your side about what you have actually done. How is the device connected to the board, what voltage level shifter board are you using (including board rev if applicable), which i2cdetect commands did you run, which I2C buses did you scan, etc, etc, etc.
I can confirm that a SSD1306 works on DB410C (96boards-sensors board driving an SSD1306 OLED display - YouTube) and the datasheet makes it look like the way SSD1309 handles multi-interfaces (parallel, SPI, I2C) looks pretty similar. However since this video is five years old at this point I’m afraid I can no longer remember how i2cdetect behaves with SSD1306.