There is also the fact that some failure modes will take both sides down. I've seen disk controllers overwrite shared LUNs, hosing both sides of the HA cluster (which is why I try to at least quiesce the DB or application so RTO/RPO in case of that failure mode is acceptable.)
HA can also be located on different points on the stack. For example, an Oracle DB server. It can be clustered on the Oracle application level (active/active or active/passive), or it can be sitting in a VMWare instance, clustered using vSphere HA, where the DB itself thinks it is a single instance, but in reality, it is sitting active/passive on two boxes.
Even if the backup stays up, failing back can be an issue. I've seen HA systems where it will happily drop to the backup node... but failing back to the primary can require a lot of downtime. For active/active setups, it can require a performance hit for resyncing.