Hello Pawel,
I just tested with the Raspberry Pi on my side
I exactly followed the instructions listed in the App. Note you have mentioned.
Here is the result:

You can see that for the first time, I also got the "NfcService Init Failed" error message as you.
So I removed the PN7160 from the socket and put it back + I checked if the I2C was enabled.
Then the NFC application started to work as you can see in the picture.
Here is the information about HW that I am using:

Also, one more thing I did at the beginning. I enabled the I2C in preferences:

BR
Tomas