This is the type of thing that I hope make into a tool like postgresqltuner, although I must confess this is the first time I have heard of it (great looking utility though!). I am more of a generalist that happens to have spent time scaling Postgres so I can't comment much on what a "true" DBA would do, but I find myself applying sensible settings in the postgres config for parameters I have used before. This would be more on the "manual" side of your question. To be fair, I think the relatively straightforward nature of PostgreSQL and the great configuration documentation makes this easy once you know a configuration parameter exists.

If I were to skew more of my development cycles to dedicated database management, I think incorporating analytic tools like postgresqltuner would be a must. Although I would probably cross-reference any suggestions with my past experience and dig into Postgres docs on the specific parameters it highlights. Regardless I suspect it would be a valuable source of additional context.