CD player

I would recommend concentrating on improving your streaming (Streamer and DAC). My streaming is as good as my CD, stored files, and analog. This is possible at many price points. A CD player is a combined transport, streamer, and DAC. If you put your money into a good streamer and DAC. Schiit makes some great DACs: Gungnir and Yggdrasil. A good streamer and you should be able to listen to a nearly infinite library (including the music you own on CDs) for almost no cost.

I have 2,000 CDs that I never listen to because I can listen to them without getting up out of my chair and often they are at higher resolution… although I seldom do because I have found so much new music to listen to.

The Cambridge AXC35 at only 350 bucks would be a great match to your ML 35XTI folded tweeter. A really nice sounding unit. 

I wanted to get a cd playing system that I thought would be good enough that I wouldn’t be upgrading from it for a while so got an Audiolab cdt 6000 and a Denafrips ares II, I’m very pleased with it. After buying the pieces, reading reviews and others' comments about them, and my own listening I think I made a good decision and am pleased with it, not looking to upgrade. My only recent comparisons are an NAD cd player that I thought was bright and fatiguing and a Schiit modi 3+ that was quite good- no sins of comission just omission, you loose some detail but the overall sound was pleasant.

Just another option,  I had over 1,000 CDs and used a Bluesound to burn them into digital files and used it to “stream” my CD library. I would never go back to playing CDs, so I agree with ghdpentice for the most part. Using the Bluesound software is so much better then a CD player. So this way I didn’t lose my great CD collection. I have now moved up to a Aurender which also has a great streamer built in. So you get to play and manage all your CDs as easy as streaming and still get a better streamer. Some Aurenders even have great DACs built in or you can get one without a DAC. You can bypass the Bluesound DAC. 
Sorry for giving you yet another solution, but it worked out very well for me to not upgrade to a better CD player (I had a Rotel by the way which was a great sounding player). 

Here we have yet another thread where someone asks about a CD player, mentions that he likes to both spin CDs and stream, and every one tells him that he should just stream.

  The OP mentions that he has thousands of CDs and listens to Classical and Jazz.

Non Classical listeners may have trouble appreciating this but music organization files do a terrible job organizing Classical Music.  I also have several thousand CDs and usually can find the desired CD on my shelves in less time than it takes to locate in the streaming software.  Making playlists, a laborious and tedious process, helps, but I have had my playlists disappear more than once due to updates in the apps.

  Now I don’t know if the issue above is germane to the OP.  Perhaps the OP has another reason to prefer to spin CDs.  It is just annoying to see a person specifying that they want to play CDs and stream,  and watch them be told that they shouldn’t play CDs.

   Now, to the OPs question…If you enjoy the DAC in the NAD but find it harsh, yet preferable to the Schiit DAC, I would replace the DAC and use the NAD as a transport.  Failing that, I would get a CD player that has digital inputs so if you later change your mind about the Schiit, you can use the CDP as a DAC for other digital sources