What are the advantages of doing database replication this way with a replicated state machine (?) vs. fault-tolerant leader election and primary-backup replication?
Availability/Scale. Paxos can expand as long as the number of participants nodes are odd (3, 5, ...), in a master-master fashion.
What are the advantages of Paxos (in particular of the variant implemented by omnipaxos) w.r.t raft?
OmniPaxos works in the presence of some network partitions where both pacos and raft fail. Cloudflare had a large outage last year where their raft based system would not gone down if they had used omnipaxos.