Time For a CD Player Upgrade

almost any moderately priced DAC will outperform the DAC in a CD player.  The DAC is the heart of the digital system so there is room for much research and choice.  I highly recommend a transport plus DAC.  

Look for a Oppo 2005 Modwright modded player or a Sony 5400 Modwright either with the Signature Truth mods.  It would have to be a used one now as Dan Wright is no longer doing the CD player mods.  He is still fixing any of his modded players if needed.  The sound is excellent in all ways and there are those of us who have done some extra things to bump the SQ up even more.  I've had Modwright Sony players since about 2010.  the ones I mentioned are the top of the line SQ wise.  If you're patient, one will come up a few times a year.  Either will be used at about $2700-$3700.  You can't do better in your price range for SQ.  

Bob

I doubt the transport part of your CD player is making anything “bright”.  I would think that would be the DAC.

have you played around with an external DAC?

@wspohn 

I've done that and compared, and there is certainly some degree of loss or error that happens when moving the data from CD to computer. The CD's still sound best. Same DAC, high end computer, etc. I personally went backwards from high-end computer/hard-wired directly to DAC to using CD's on a McIntosh MCT500. 

The most natural sounding CD player I have heard is the Ayre CX-8.  A bit above your stated budget new, but someone with audiophilia nervosa may put one on the used market…