SQLite as an embedded server-side database requires extra work and configuration to make it a viable alternative to PostgreSQL. It lacks good write concurrency and recoverability by default. It is, however, continually improving but gaps remain.
Since it does not have a wire protocol, SQLite is rarely connected to a data warehouse via ETL so it does not fit well as an alternative to TiDB.
No, it depends on the server application. An all-read or read-mostly server requires nothing special. Same for any server that expects a low number of users or database per user(s).
Mozilla uses it on servers for documentation sites. You can also run your own Firefox sync server using SQLite.
It has the advantage of requiring zero administration. All you need is a file: backups or copies are just a matter of copying the file.