You would be lying if you wrote that because you do not know if that is true.
But that's not true; it could easily fallback to other forms of geolocation like using the current IP.
Good lord. So could literally any app on the planet
That would allow you to see the local network IP (not actually sure you even get that, tbh). To get more detailed information about IP configuration, you need Location permission. Been there, done that. Most Android network information calls provide degraded information if you have not been granted Location permissions.
If an app can make an HTTP request, the app can know the user's public IP address and the geolocation derived from that.
This data has well-known limitations, but I think it is the fallback people are talking about here.