There are definitely players that can't play CDR / CDRW based on the transport in them. As Buckingham points out, there are also instances where some will play, some won't. I have used a Sony DVD player as a transport for quite some time and regularly play CDRs on it - never a problem.
The Sony XA777ES suggestion above is certainly a good one for all the reasons listed. You should be able to get one brand new for about your target price (try www.oade.com). The latest rev. of the Muse Model 9 can play those formats as well as MP3. It's a great sounding player and doubles as a DVD player if that's of interest. It doesn't do SACD which, if that appeals to you, probably makes the Sony a better choice. The Muse is also more expensive - for a used Model 9 Signature, you're looking at $2500-2800. Money well spent in my opinion, but more than you're looking to spend possibly. -Kirk