Publish to Web Public URL Filtering

Without seeing the report builder behind your example, it's difficult to say for certain, but my guess would be the Country name you are passing isn't a filter but is instead a "page" name in the report. It looks like each country in this example report has it's own page, but you aren't applying a filter through the URL. So this method won't work in your case unless you also separate the different cases out to different pages.

Instead, you could try using this filter format on the URLs: Power BI URL Filters made easy - PowerPlatformLearn | Rory Neary

Which should be basically something like this:
https://www.exampleurl.com/$reportid=fjdfksd-fsdkffjs-dfjsdfksjdf&$filter=TABLENAME/COLUMNNAME in ('Value1','Value 2')