There is an existing f-droid version of Maps.me [1] and this looks like a separate effort. Despite having access to the source code, map hosting is a non-trivial problem [2].
It becomes trvial if one uses MapTiler or Mapbox with their own styles and overlays. And even self hosting has much improved since the days of raster tiles.
For downloadable maps you need vector tiles and a mobile rendering engine like OSMand and other mobile apps use. MapgoxGL maybe, but that one uses their service, I don't know if you can serve local data. You could with a local webserver.
You can use openmaptiles to create the vector tiles (it is made by the Maptiler guys) and serve the tiles with tileserver-gl (or tileserver-gl-light, if you don't need raster tiles).
The fork has maintained their own tiles, so I don't think there's any question of not knowing how to. It's just been inconsistently maintained over time.
With openmaptiles, they had a hiccup when Mapbox sent C&D, because they used the same mvt scheme as Mapbox (thus they were able to use the same themes). Once they switched to their own scheme, I didn't notice any problems with maintenance.
With tileserver, it is more interesting. The full one (capable of generating raster tiles on demand) still needs ancient node.js (v10, soon to be eol), which can be a problem. The light one runs OK on the current stack.
We pay something like 35$ per month for four servers and the cheapest plan. That's peanuts compared to the infra we'd have to maintain ourselves and work we'd have to do to build the tiles or money to spend on a yearly tile subscription.