Would it be a good idea of Duplicati would also put the local database (encrypted) on the remote location?

This is not about the files you back up. This is about Duplicati’s configuration. In my case the contents of~root/.config/Duplicati/ on my Mac.

In the scenario that I sketched, I was unable to repair or rebuild from the location where my list and block files are stored. Duplicati kept complaining. This was when I encountered it. Missing remote file was the complaint when trying repair or recreate.