Upgrade my CD player or futile effort?

Whether DAC or new player the question is what makes them sound better than your modest unit?
Answer: Internal analog amplification circuit, power supply and filtering capacitors. Look under the hood.
I have a Marantz SA8005 SACD player and if you look for online pics of the chassis you will see what I mean.
The analog amplification circuitry gives the sound its drive, dynamics and energy as well as freedom from noise. The SA8005 does this quite well but is out of production.
The newer Maranttz SACD players do this well too.
I also have a Luxman D05 which uses an R core transformer power supply, the best IMHO. Lower noise and improved dynamics.
As others have mentioned The D03X is an excellent sounding player.
I prefer a one box solution. However there are some excellent DACs with robust power supplies. Look under the hood at online pics.  Avoid DACs or players with empty chassis.  

Hello,
The Ares is a nice DAC but I would stick with a chip based DAC. The R2R can sound a little warm or unresolved with your system. You might want to try the RME. The reason is you can set up the sound to your preferences. It will be slightly less musical than a Chord. But a good starting point into the DAC world. Plus it comes with a remote. If you feel this is for you they have a black version that steps it up a notch. I hope this helps. 

Budget... new or used?

A dedicated CD transport and an external DAC is definitely the way to go.  I endorse the Audiolab 6000 CDT as a very good transport; there are lots of really excellent DACs.  You don't have to spend more than $2000-3000 total to have something approaching state-of-the-art CD reproduction (note that I said "approaching" - not "reaching").  I am sure you will be a lot happier with CD sound than you are now.

I went with the RME ADI-2-DAC fs that hshifi mentions and love it. It has more features surrounding the actual DAC than the competitors, but that might be more than you want to deal with. I am in total agreement with the above advice that you change one thing only to start with. That would be the DAC. Keep it separate from the transport player so you can upgrade either as you desire in the future. I even think the Topping Dx7 Pro would improve your system and it would cost a lot less to try. Buy from a place where you can return it, no questions asked, if you don't think it is a major improvement or to your liking. Plus, it comes with a nice headphone amp, so you can find more places to spend your money in the future. :)I have ripped all my CDs and find I rarely listen to either the actual CD or the ripped version. I started with Qobuz and thought it was great, then tried Amazon Music HD and never looked back. It sounds great and it probably has any CD you have plus tons more. It has a great interface to pick an artist and it lists all their hits to see if there are ones you like. It also has so much artist and album information to get lost in for...you pick the timeframe.  Don't get me wrong, I am not a Jeff Besos' fan, but there is a reason why they are the largest at what they choose to do. Just my opinion, but I chose to not buy into the MQA/artist platform (Tidal).Have fun!