"near-Byzantine" is not a very clear term you can reason about. A system is either Byzantine-fault-tolerant, in which case it handles all Bizantine faults, or it is not. A system that is tolerant to some faults (that you may want to call "Byzantine") is not BFT.
You don't call plaintext SMS "tamper-resistent" because it resists to some simple tampering. You don't call your house "FBI resistant" because you managed to convince them once to turn around.
A Byzantine fault is clearly defined as a case where a specific node may be doing anything, including not know it has failed, including malicious behavior. It is important that people know what class of faults their system is designed to resist; for Raft/Paxos, it is NOT Byzantine faults. Those systems are pretty great, but trying to pretend they aim at BFT is dangerous misinformation...
What then would you specify as the clearly defined storage fault model for non-Byzantine protocols such as Paxos/RAFT that rely on stable storage for correctness?
Anything is possible with Byzantine faults, on the specific failed node. It will not remember voting, it will not remember to vote, it will not remember its identity, etc. PAXOS/Raft are not tolerant to a minority of nodes exhibiting those kinds of faults, only to a minority of nodes being unreachable or partitioned.
Remember that the Byzantine generals had traitors among them, not merely communication issues.
You don't call plaintext SMS "tamper-resistent" because it resists to some simple tampering. You don't call your house "FBI resistant" because you managed to convince them once to turn around.
A Byzantine fault is clearly defined as a case where a specific node may be doing anything, including not know it has failed, including malicious behavior. It is important that people know what class of faults their system is designed to resist; for Raft/Paxos, it is NOT Byzantine faults. Those systems are pretty great, but trying to pretend they aim at BFT is dangerous misinformation...