Why isn't 'classic' Multi-Paxos Byzantine Consensus used for public Blockchains?
Nakamoto Consensus has some properties that are superior to BFT algorithms. (It also has some properties that are inferior, but that's another story). 

First, BFT algorithms assume a central authority picks who the participants are. NC, on the other hand, allows anyone to attempt to win the next round.

Second, BFT algorithms generally require 2/3 + 1 (e.g. 67%) of the participants are honest. NC sets a lower threshold of 1/2 + 1 (e.g. 51%). That's because NC's PoW prevents a node from broadcasting two messages at once. 

Finally, NC has the interesting attribute that the moment a miner "wins", they also simultaneously choose the transactions they are going to include on the block. This means there is no window of opportunity for an attacker to attempt to influence the winning miner to break the rules. 

Note that Paxos is not a BFT algorithm. Paxos solves for stop-fail failures. The worst a node can do is stop working. Paxos does not solve for "byzantine" failures, where a node can try to say two different things.