That's the radio I was using until a few weeks ago, when I got my Sangean ATS-909X. Somehow the Sangean manages to outperform the ICOM, although the ICOM admittedly has better IF filters and a nice, smooth tuning dial.
Anyway, you're right: Internet radio is not the same. I have a few .M3U playlist files that allow me to hear the streams from online broadcasters on my media player without going on the web. However, they include only a handful of international services in English; namely Radio Poland, Radio Romania International, Radio Ukraine International and the BBC's World Service. The rest of them are domestic services complete with commercials and usually in foreign languages, of which I can only understand German. I ignore the languages I can't understand and listen to those stations only for the music.
International broadcasters were what attracted me to shortwave radio many years ago. Now I tend to fill in the gap by decoding some digital transmissions, but they keep me tied to this infernal computer!