You are unfairly associating the inconvenience of "tons of small databases" with SQLite only because SQLite is good enough to make such an architecture practica

You are unfairly associating the inconvenience of "tons of small databases" with SQLite only because SQLite is good enough to make such an architecture practically plausible; it shouldn't be assumed that it is a good architecture in the first place, or that problems are SQLite's fault. If you start having trouble "beyond customer #5", you have a problem of labor-intensive manual changes and insufficient automation, not a task that a different DBMS could do better.