DVD playback worked on older models as the DVD drives they had featured an undocumented function for 3x playback (normal Wii read speed was 6x playback) which many have theorised was for when Nintendo introduced the DVD playback functionality* (they had mentioned it could happen at an earlier e3). For various reasons this never came to pass and presumably as people were using the feature to play copied games the feature was dropped from later drive revisions. You probably have one of these later drive revisions.
Short of replacing the drive there is not a lot you can do. You could rip it and stick it on a hard drive I guess to use with the media player but if you are doing that you might as well rip it and encode it in something smaller.
*when doing things in the commercial world (or at least in the US and probably Japan) you have to pay your dues to the MPEG-LA and more besides if you want DVD playback-- the original xbox only allowed it with a remote for similar reasons and MS have stripped the DVD functionality from their newer OSes for much the same reason.
