Earlier quoted context omitted.
The point was merely that it cannot be used to prove "anything" since the actual conclusions do not necessarily follow, leaving the original claim that you can prove anything with false axioms unproven.
The conclusion "Bertrand Russell is the Pope" does necessarily follow from 1=0. I'm not sure why you think it doesn't. The conclusion "Bertrand Russell is not the Pope" also follows. Yes, this is a contradiction. So is 1=0.
The claim was that if you accept a false premise, you can prove any false thing to be true. What we've actually seen is random claiming of contradictory things that supposedly follow by people trying to appear clever. It so obviously doesn't follow that it's a little strange we are having this conversation. If the conclusions themselves can be seen to be contradictory, anyone can dismantle your proof and show that it's bogus, even if you do accept the original premise. So the claim is bogus, and it's probably not a real story anyway.