You could of course do that, but the main reason to use SQLite in a server application is for the performance characteristics of staying in-process for accessing the database. This makes it cheap to make queries in response to a previous query, which would need to somehow be batched for good latency with a client/server database. Re-adding the client server model defeats almost all of the PROs of SQLite, leaving you with only the CONs.

Traditional client/server databases let you run app logic local to the storage. E.g. stored procedures. There's no particular advantage for sqlite there.

Now, people add unnecessary layers all the time to everything. But there are also good reasons for compute that is between "local to the user" and "local to the storage".