No, the issue is it doesn't have high availability features: failover, snapshots, concurrent backups, etc. (Edit: oops, comment pointed out it does have concurrent backups.)
SQLite isn't a toy DBMS, it's an extremely capable embedded DBMS. An embedded DBMS is geared towards serving a single purpose-built client, which is great for a desktop application that wants a reliable way to store user data.
Once you have multiple clients being developed and running concurrently, and you have production data (customer accounts that are effectively legal documents that must be preserved at all times) you want that DBMS to be an independent component. It's not principally about the concurrent performance, rather it's the administrative tasks.
That requires a level of configuration and control that is contrary to the mission of SQLite to be embedded. They don't, and shouldn't, add that kind of functionality.