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