Managing large numbers of raster DEMs for QGIS

I have a large number of (say 9000) 2km x 2km raster DEMs in ASC file format that I am trying to manage, in order to use in QGIS.

I’ve been looking for way of storing them, ideally on the filesystem, but in a way that they can be searched/queried geographically. Also, so that they can be added to over time. Then loading the DEM data into QGIS as necessary to do ad hoc processing, usually around 10-20 at a time.

Is there a way of doing this using say GeoPackage format, or any other method readable by QGIS?

The files could be converted to GeoTIFF if that makes any difference.

Also, if there’s no way of storing them on the filesystem, but there is a way of storing the data in the file, that would be helpful. I have stored one file in a GeoPackage, but trying to then add additional files ends up with the original being overwritten.

I have also had limited success in storing data in a Rasterlite (SpatiaLite) DB. While QGIS will load the data, it doesn’t seem to have an option to select what area of the data is loaded, and this would seem to be impractical for a large data source when you only want to operate on a small part of it.