"FATAL: the database system is shutting down" while creating connection to PostgreSQL

This issue turned out to be caused by a very bad misuse of PostgreSQL where our server had booted two PostgreSQL instances on the same data directory (it had deleted postmaster.pid and used a new port number, so the normal safeguards against this behavior were bypassed) which explains why the logs did not hold any useful information -- they were being overwritten by the PostgreSQL instance which wasn't refusing connections. The actual issue was caused by a complex interaction between the competing PostgreSQL instances, and I hope nobody else runs into this issue as well!