It’s not latency, it’s throughput. Broadcast and incast network communication patterns in the LAN kill throughput, and are present in any quorum protocol. Atomic broadcast protocols like chain replication or LCR which use strictly sequential communication patterns (via chain or ring topologies) have much better throughput in LAN environments than Paxos/Raft (except for Ring Paxos, which requires IP multicast). In addition, because they outsource group membership and failure detection to a separate (likely Paxos-based) service, they require only f+1 members to tolerate f faults, rather than 2f+1 (see Lamport et al’s Vertical Paxos work). But they have two main downsides relative to Paxos/Raft-like protocols: latency grows linearly with cluster size and is bounded below by the slowest node (so they aren’t a good fit for high-latency WANs), and as noted above, they can’t “self-host” like Paxos/Raft.
PS Apparently my previous comment outraged the HN hive mind, judging by downvotes. I find this hilarious and revealing given that all I’ve said is obvious to any actual distsys researcher (ask eg the HyperDex or FawnKV folks about why they chose CR instead of Paxos). And at one large cloud provider I worked at, this was taken for granted back in 2012. Apparently “distsys” pop culture has yet to catch up.
I should also mention that a great virtue of quorum-based protocols (for majority or basic read/write quorum systems) is their ability to mask tail latency/transient faults. That is difficult to replicate in sequential protocols and is a good reason to use quorum protocols in high/variable-latency and otherwise unreliable environments like geo-distributed databases.
I’m not sure how often Fast Paxos is really a good tradeoff, since in the LAN it has inferior throughput to the other protocols I mention below, while in the WAN >3/4 quorums expose you to additional tail latency, probably enough to negate the advantage of eliminating a single message delay if clients are distant from the cluster (see the Google SRE book for discussion).