In my own tree, I tend to accept the GeoNames coordinates from the Place Cleanup addon gramplet, and fallback to Wikipedia coordinates.
GeoNames says that it tries to use the centroid of the bounding outline.
Then plot all the Places enclosed by a particular Administrative region (like a Country or Province) with a Filter in the All Known Places geography view mode. (It helps to View → Configure the “places marker color” for that map. You can drag swatches instead of using the Select button to pick colors for each type one-by-one)
When exploring All Known Places map pins, any plotted outside a reasonable location can be tweaked. For instance, I move the USA country pin to just off the Atlantic coast of Washington DC.
In the AI experiment above, I did not demand that Perplexity use a particular source or method. So who knows what criteria was used.