SQLite already allows multiple connections, so putting it on a server and adding a program that talks a network protocol and proxies the queries to the DB sounds more logical to me?
Because I can use SQLite and its "a file is all you need" approach as long as I don't need multiple machines.
And only bring in the other software (the proxy) when I need it.
Writing a networked application that uses SQLite as a database is perfectly reasonable. You're just making the decision to lift the layer of abstraction that is concerned with machines from the DB to your application, which may or may not be a reasonable thing to do.