This is a sort of broad question, but if I were to try and answer I would say that if there is anything you want to be handled by the database you would gravitate towards an extension.

PostGIS is a great example since one of the most well known it additions it makes is a datatype, but that's not the only thing it adds. A datatype on it's own wouldn't be that useful without Spatial Indexing (https://postgis.net/workshops/postgis-intro/indexing.html) or spatial utilities to make spatial joins more expressive (https://postgis.net/workshops/postgis-intro/joins_exercises....).

Rich data types aren't the only candidates for extensions, automated partition management (https://github.com/pgpartman/pg_partman), data sharding (https://www.citusdata.com/), or even database cron scheduling (https://github.com/citusdata/pg_cron) are also good examples of things that are well suited to be extensions.

To be honest, I have a hard time imagining alternative paths where some of this functionality _isn't_ an extension. You might imagine an external daemon for things like pg_cron and pg_partman, but if PostGIS wasn't an extension you would probably be using a different database/tool (or a fork) if you had geospatial requirements. It's worth noting that Citus was a fork of PostgreSQL before it was refactored to be an extension.

I know this isn't a direct answer to your question, but hopefully you find it somewhat useful.

Edit: I totally forgot to add 1 more thing about extensions. When functionality is packaged as an extension there is a much higher likelihood you can mix and match them. For example, I currently run a PostgreSQL cluster using Citus, with pg_partman for automatic time-based partitioning, pg_cron regularly scheduling partition creation, and PostGIS for geospatial datatypes. You could extrapolate the various ways you might use this kind of setup...one that jumps to my mind is a scalable GPS history that could be queried by time period as well as by geographic region.