So the usecase is: I started with SQLite, but now I have too many terrabytes to fit on one server? That seems.. very uncommon.

And since moving it out of process, and even to another network, is going to make it much much much slower. You're going to need a rewrite anyway

I think it's more like you started with SQLite and now you need concurrent writes, replication, sharding, etc. etc. - all the stuff that the "big" databases like PostgreSQL provide.