There was a problem I didn't mention with the control panel, I couldn't control it anymore when the Arduino was not powered.
This could probably have been fixed by adding diodes but as I will need to power the Arduino anyway if I want to use Bluetooth, I just chose the later.

So I just soldered a pin on the on an unpopulated part of the PCB that said "5V". I first tried one saying "5V Backlight" but it turned out to be 0V, it is probably activated by the MCU on other models of treadmill.
Then I connected the Vin of my Arduino Nano to this pin. From what I read, 5V is not enough to power the Nano through Vin, you must directly connect it to the 5V Pin (which makes sense but I thought it was an output only pin).

Also I unsoldered the buzzer because it was too loud. I soldered pins instead so may be I will add a resistor or feed the signal to the Arduino. It feels a less safe to not have the three warning beeps before the treadmill starts, I will try to find something safe and not annoying.