As you pointed out, really not the same use case.
Firefox and Chrome use SQLite. Imagine if you had to download and install PostgreSQL just to browse the Internet.
SQLite is not meant to be a multi-user shared database. It's never going to replace a database server if that's what you need.
The reverse of your question is: is there a good reason to use PostgreSQL when you can use SQLite?
Yes but this article is about web services where you clearly can use SQLite but I wanted to understand if there were good reasons to give up Postgres' advantages to do so.
SQLite is not meant to be a multi-user shared database. It's never going to replace a database server if that's what you need.
The reverse of your question is: is there a good reason to use PostgreSQL when you can use SQLite?