I always have our appliance plugged in, because when it is freshly started, you always have to press the button on the front to confirm the setting.
If you can memorise the setting (press 2 or 3 times), it's no problem.

You can switch on "auto power on" in the app. The device then starts as soon as it is supplied with power via USB. It then automatically connects to my smartphone.

I'm not sure what you mean. Does the device power on automatically when you plug it in? - No you have to press the leftmost button on the front panel for a second or 2. Does it select DAC automatically? No, you need to manually select it using on of 3 mode settings on the front panel. I know the manual is not great but I would recommend re-rereading it.

Hello
So you have three options.

1. input with Bluetooth and output 2 x cinch stereo to amplifier.

2. output Bluetooth to headphones or DAC with Bluetooth input. The source can be Toslink or coax.

3. input USB from PC or Toslink or Coax to Cinch output.

For points 1-2 you need a USB C cable for the power.
For point 3, the power comes from the PC/laptop.

When I was once in the Fiio menu via Bluetooth input with the Fiio Control app, I only saw options such as fixed volume and others.

But I am not aware that the DAC is switched on directly with signal input.

In any case, you have to press the left button briefly and hold it down a little longer to switch it off!

That's it!