So, for example, typing the country name could select the country.

Or the field is empty (like on the main location search) and when you start typing, it suggests countries that match.

Or repeatedly typing the first letter of the country name would cycle through countries starting with that letter (e.g. typing C would get Cambodia, then hitting C again would get Cameroon, then Canada, etc)

The country names could also have their flags next to them, to make them easier to find.