Do I need to move off SQLite now? 3k users, one server

You're not being stupid. A few hundred writes a day on one server is a tiny load for SQLite. Postgres wouldn't make it faster for you.

But one thing needs fixing now: cp of a live SQLite file can produce a corrupt backup if a write happens during the copy (especially in WAL mode, where recent data is in the -wal file). Use the backup API instead:

sqlite3 app.db ".backup '/backups/app-$(date +%F).db'"

or VACUUM INTO, or Litestream for continuous backups to S3-compatible storage. Test a restore once.

The real reasons to move later: you need a second app server, or several processes writing heavily, or you want managed backups and point-in-time restore without running them yourself.

I mostly agree with Fatima, with one push back: medical practices tend to ask about backups, restore time and where data lives. A managed Postgres gives you point-in-time recovery and a nice answer for that questionnaire. Not a performance reason, a sales reason.

Tell Claude Code in your CLAUDE.md that SQLite is a deliberate choice. It will stop suggesting the migration every time.