The bitmap approach you describe allows for immediate (i.e. O(1) ) lookup of region by coordinate, which is pretty neat. Space-efficiency-wise, a bitmap (+ index that maps color to country) might not be the most efficient data structure, though, as there are more than 256 countries, so you already need 16 bits for each pixel instead of 8. Then, you have the additional complexity of if you actually want the bitmap to be viewable by humans, you need to make sure that the colors for neighbouring countries at least are sufficiently distinct.
Anyway, a Kotlin library I wrote uses a similar technique to make requests for the majority of locations immediate, while also handling the edge cases - i.e. when querying a location near a border.
https://github.com/westnordost/countryboundaries (also available in Rust)
What it does is to slice up the input geometry (e.g. a GeoJson) into many small cells in a raster. So, when querying for a location, one doesn't need to do point-in-polygon checks for potentially huge polygons, but just for those little slices that are in the cell one is querying for. And of course, if a country completely covers a cell, we don't even need to do any point-in-polygon check anymore. All this slicing is done in a preprocessing step, so the actual library consumes a serialized data structure that is already in this sliced-up format.
I needed it to be fast because in my app I display a lot of POIs on the map for which there is logic that is dependent on in which country/state the POI is located.
There are 249 countries with an iso code; so 8 bits might be enough. So it's not that bad. But even at 32 bits it would probably be fine and you could cram in some more data.
There are many similar things of course but nothing that was multiplatform, which I needed. I actually created a multiplatform kotlin library for working with language and country codes a few months ago: https://github.com/jillesvangurp/ko-iso
It seems we have some shared interests. I'll check out your library.
What you describe is nice strategy for indexing things. I've done some similar things. Another library (jillesvangurp/geogeometry) I maintain allows you to figure out which map tiles cover a polygon cover a polygon. Map tiles are nice because they are basically quad tree paths. I have a similar algorithm that does that with geohashes. You could use both for indexing geospatial stuff.
Slicing up the polygons sounds interesting. I've been meaning to have a go at intersect/union type operations on geometries. I added a boolean intersects recently to check whether geometries intersect each other. I already had containment check.