In the very first paragraph of the OP I explain why I use SQLite and PostgreSQL is an excellent example of why I suggest people start with SQLite.
To suggest using PostgreSQL (or MySQL, or any of myriad other database engines) is a simple as replacing the connect string with a different connect string amounts to deliberate infliction of pain and suffering. None of the non-"toy" databases are trivial to install and get going. I have no experience with PostgreSQL, but I bet it is little different to install and configure than MySQL, and that can be a right royal PITA.
The intent of the OP was to provide an easy introduction to database use and SQLite, with its trivial install and very easy configuration (i.e. none), is excellent for the purpose. Toys have their place and if SQLite is a toy, then it is an superb one where most of its widgets and gadgets work just like the real thing that grown-ups use.
People almost always start learning on a toy or reduced functionality version of anything to reduce the barrier to learning. Should we make it harder to learn to use a database by requiring extra layers of set up before you can write a single line of code just so you can say you are using a professional strength version?