I have the Marantz CD6007 paired with a Chord Qutest DAC & am still not happy with my CD sound. It is infinitely better than using the internal Marantz DAC, but doesn’t compare to streaming or vinyl. So while a DAC upgrade is a definite improvement, I am not sure it will get you where you want to go.
Perhaps a lower end Topping DAC paired with the Audiolab 6000CDT would be a great solution well within your budget… very similar to @audphile1 recommendation.
Audphile1/others: which Audiolab do you recommend if I were to upgrade? I am a very infrequent CD listener, but that could change if I could get CD sound on par with vinyl and streaming. Or perhaps I just load all my CD’s to my Aurender and call it a day.