I also find it interesting that QF requires the paper printout - I have used mobile BPs on numerous airlines both in Australia and elsewhere, and QF are unique in doing this IME. I have two theories about why QF require it:
1. There was (perhaps still is?) a rule that pax turn off their phones if boarding via the tarmac/stairs. If people complied with that, there would then be delays at the aircraft door while people turned their phones back on before finding the mobile BP, so perhaps the paper slip was/is intended to avoid that issue.
2. As far as I can tell, if you check in and get a mobile BP, and your seat assignment changes, the QF system does not seem to be capable of updating the mobile BP.
BTW, I once had a really weird experience with this. I was boarding a QF domestic flight with a mobile BP and got the paper slip as usual. On reaching the aircraft I went to take my assigned seat, only to find someone else in it. I politely said that I thought they were in my seat, and showed the BP on my phone. They showed me their (conventional hard copy) BP showing the same seat number. I then checked the paper slip I was given, which showed a different seat number - which as it happened was in J! Presumably I had been given an op-up but didn't know about it - I don't recall anything being said by the gate agent to alert me to this, but it was about 6am so perhaps I was just half asleep!