Why would you want to use PostgreSQL in the contexts that SQLite is used?

SQLite is fantastic when you need a data store for mobile or embedded use, you'd like ad-hoc queries over your data with the option of adding indexes to speed up frequently used ones, you want transactional update to the data, and multi-user access is not a problem.

It's basically a much better replacement for the hash table on a disk approach of DBM libraries.

PostgreSQL would just be total overkill for this kind of scenario.