You might be able to compare some things and repair the 'old' board after all.
It would be great but for now I just hope that replacing the board will fix the mixer. I don't want to risk damaging the new board right away. Maybe after some time, when I get over the price I had to pay

Does a brand new mixer cost more than that shipment cost? 
Fortunately yes, these mixers are pretty expensive and cost around 750 EUR.
Did the micro really draw 57 mA? I think that measurement is not correct because I'm sure the capacitive power supply can't supply that kind of current.
What has probably failed on this board is the capacitor in the capacitive power supply. From the pictures of your board I think that is C11.
https://ww1.microchip.com/downloads/en/appnotes/00954a.pdf
I have a decent power supply and it was showing 57 mA when I injected 5V in paralel to C8. Like I wrote in earlier post I was able to disturb the microcontroller function by touching XTAL1 (PIN7) with multimeter or oscilloscope probe. Then the current dropped to 5 mA. So it looks like microcontroller draws about 52 mA.
But you are right. This current is too big for this kind of capacitive power supply. I saw it also in my LTspice simulation that everything above 20 mA will lower the output voltage of +5V supply rail. The capacitor that is part of capacitive power supply is the C9 = 1 uF (btw. in simulation I missread this value and placed 100 nF) '105 (K) X2'. The C11 is in paralel to AC input voltage and doesn't play any major role in current efficiency of that power supply.
However I would expect that 57 mA will break down the 5V supply rail... but when I probed the board under AC I measured that the micro had a stable 5 V going into it.
2. The PCB board is supplied by AC and ATtiny861 on the board is supplied with +5V. But I dont know how to diagnose if it's alive or not.
By the way thank you for that application note really explains the topic nicely.