BD players make excellent CD players. If you get one with GraceNote (like Sony, LG and a few others), it will also show track names and album art. However, if you are a musician who does analysis or transcription, you'll find a CD player is better because you can set A-B repeat segments, change playback speed, etc. So it depends on your requirements. I use a PC for that kind of thing anyway, so a BD player still works fine for me. One other thing to consider is the type of CD's you have. Only certain players can handle formats like SACD and DVD-A. I doubt your current CD player handles those anyway.
But these days, it's more convenient to rip all your CD's to a computer and stream your music via DLNA instead of shuffling discs, IMO. That way you can search/sort by artist, album, title, genre, etc., and make any number of folders or play lists. Most BD players can do that as well.