Is geonames the best data source for finding geographic place names for every place on earth? Or is there any open data that comes closer to the level of Google or Apple maps? Just looking for place names, with potentially geographic coordinates, nothing too fancy.
Geonames does not only include cities, but other features too.
In the tab-limited file, there are fields (columns) for feature class and feature code. http://www.geonames.org/export/codes.html
For example the class with a code "T" (for mountains, hills and rocks) includes a feature with a code "ATOL" (for atolls).
Why not to look on OpenStreetMap data? You can use Nominatim for search, https://nominatim.org/release-docs/latest/api/Search/, you can use Overpass for spatial data queries, like this one for sumits: https://overpass-turbo.eu/s/1fCT. There is a world outside Google and Apple, very often much better mapped.
Maxmind has a world cities database too that has a list of cities and countries with geographic coordinates