SQLite is a great embedded database, but is deliberately not designed for replacing a 'full' database server. It doesn't support highly concurrent usage, it can't be run as a network service, it has a comparatively weak type model.

All of these differences are actually assets for the embedded DB market. They could be fixed, but you would end up with a database that was a winner in neither space.

I think the killer problem with SQLite for businesses is that it essentially locks their data inside the application. With a full SQL server, the data is trivially exposed for use / integration with other systems.