How to import an OBF map file into OsmAnd?
To view OBFs that I didn't download through OsmAnd, I first had to change the directory where its data is stored to one where I can push stuff (non-rooted phone).

For that, I first went to Settings > General settings > [Miscellaneous] Data storage folder and switched it to "Manually specified...". I decided it would be /storage/emulated/0/osmand as this is a place I can see when connecting the phone to my laptop. OsmAnd asked whether it should move all its config files there, I said sure, go ahead.

I put the OBFs directly at the root of /storage/emulated/0/osmand. It lost all the maps I had downloaded through the app after the folder change, so I had to download the world map again, for example, and it ended up at the root of the folder too, that's how I figured it out.

It may take an OsmAnd kill and restart for it to detect new maps, and the trick above of renaming the file through the map manager indeed made it be inspected by OsmAnd and marked in green as a region for which a detailed map was available. Even before the renaming, its content would still be used though, so as soon as you see something corresponding to your OBF in the map manager, everything should be OK.