I had a very similar problem with my old installation, and reinstalling the game with a fresh version fixed it for me. My server list was also empty, even though my internet connection was working normally. I ended up downloading a new copy from https://cs-unikov.net/ and after installing it, the server browser started working again. I could see available servers and connect to them without any additional configuration. If your current CS 1.6 installation suddenly stopped finding servers, replacing it with a fresh version is definitely worth trying before spending a lot of time looking for complicated fixes.
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