Well, there's Turso and Cloudflare D2 as examples of how you can scale SQLite in practice.
In terms of a simple API over SQLite, it will depend on the number of users, the overhead and the cache ability of tall data in practice.
If you have a heavy transactional load, it may not go so well. Similarly, any service may be better implemented in a single connection with seriously requests, which will limit this but reduce complexity.
Realistically, PostgreSQL in Docker is pretty much my baseline for a shared database server. If other systems really need to access it, it's going to be the easier path.
Even Firebird would likely be a better option for many lower end use cases where a shared service is needed.