Earlier quoted context omitted.
The haproxy method doesn't rely on DNS at all so I'm a bit confused.
well either it uses DNS for failover or you have ipvs (lvs, keepalived) enabled or worse if the machine with haproxy crashes your basically dead. Of course there is also bgp and anycast, but this is not "cheap"
That's not how it works. It is very similar to sidecar approach which various service discovery solutions have. You have haproxy running locally on localhost and you communicate with it, the haproxy then routes the request to right nodes. No DNS, no LVS and no keepalived.