If m3u and asx aren't listed, than that right there is a problem, because m3u and asx playlist types are exactly what I use on NWR to ensure streams are opened in appropriate players. ASX is a windows media playlist, m3u is a Shoutcast, or WinAmp playlist, windows support for the latter is pretty weak.
Windows Media support for SC PLS streaming is extremely limitted, so the ASX generally points directly to the mp3. If the mp3 is encoded in a format that WIndows Media doesn't support, you are screwed. Windows Media Player does NOT support every codec available, that being said, there really isn't a single player that does, but many more available that support further codecs. The M3U type points towards an SC PLS stream, which typically modifies the encoded type slightly, and is probably allowing you to listen. WMP is probably opening these files as a default, and thus you're having some success with it.
Different people have different configurations, there is no one solution that works for everybody, it's best not to suggest an errant configuration such as yours however. If what you have set up works for you now, then leave it that way. There's billions of possible configurations, stick with what works, until it doesn't work.
The best suggestion I have for this case is that the artist uses a more standard codec for the mp3, thus allowing more compatibility.
W.