How to monitor general performance and wellbeing of a database?
In general I agree somewhat: if the customers are happy, don't start tuning.
But there are certainly things that can go wrong without you noticing before it is too late. Here is a random list of things that should cause an alert, without a claim to completeness:

Is there enough free disk space?
Is the CPU overloaded?
Is the I/O subsystem overloaded?
Does the database grow unreasonably?
Are there problems with WAL archiving?
Are there too many client connections?
Are there long running queries?
Are there transactions that remain open for a long time?
Are there databases that are in danger of suffering data loss from transaction wraparound?
Can you connect to the database and run a simple query?
Is replication delayed too much?
Are there any tables with too many dead tuples?
Did you get errors from data corruption?
Are there any stale prepared transactions?

pgbench is more useful than you think: you can use it with custom scripts that even support a simple scripting language to simulate different workloads.
But of course it is essentially a tool to generate artificial load to test the performance impact of certain changes and to make sure you didn't misconfigure PostgreSQL or the operating system.