Write a method that fetches country data from the REST Countries API and processes it into a Map grouped by region. Make an HTTP GET request to https://restcountries.com/v3.1/all and return a Map where keys are region names and values are lists of country names.

Handle network failures gracefully by returning an empty Map. The method should parse the JSON response and extract country names and regions.

Examples:

Map<String, List<String>> regions = processCountriesFromAPI();
// Returns: {'Europe' => ['France', 'Germany', ...], 'Asia' => ['China', 'Japan', ...], ...}

Map<String, List<String>> emptyMap = processCountriesFromAPI(); // if API fails
// Returns: {}