Yep, one full copy of the DB on each app server. So yes, it's bad if your data is 100s of GB, but if you only have in the low 10s GB of data or less then it's fine. It all depends upon how many server you need to run your app, If you only have one or two server then even very large databases of 100s of GB could be just fine.
Interestingly, our SRE people specifically have a rule "never share the DB and the app on the same server", because apparently a high load on the DB can decrease the overall performance of the server. It also allows them to freely add new DB servers (sharding) without touching the app server
That makes sense with client/server databases and if you are trying to separately manage database resources from app resources. When you use SQLite you have basically decided that the database and the app is one integrated unit that you will manage as a whole.