However, that Blu-ray player has no front panel control, no front panel display, everything is done via the remote so I can't use it in my sound system since I would have to be able to tell what track is playing.
Would a cheap Walmart DVD player with audio output solve this problem? Assuming DVD player can play whatever that the CD can play and more.
Yes, replacing a 25 year old worn out broken CD player with a new CD/DVD player will solve your problem, just like replacing my 20 year old Sears Kenmore washer solved my problem.
But, unlike my new washer, a new DVD player, even though affordable, will also make new problems.
You discovered one problem during your problem solving journey: navigation. Depending on the DVD player you may or may not get a track number, which is real hard if you want to hear just one song. Don't even think about audio books on CD, finding the chapter where you stopped is an exercise in counting.
A second potential problem is audio quality. DVD players are optimized for coax digital output, not analog. You may have noticed I used the digital output of my cheap Sony DVD into an AVA DAC. I never tried the analog outs. I would guess that low jitter, noise rejection, filtering, multiple regulated power supplies, dynamic range, transparency, polypropylene capacitors in the signal path and the newest high-capacity power supply capacitors are not included in the analog output of a $35 DVD player.
What did you pay for your CD player 25 years ago? You should spend at least that much now to get the same sound quality now, probably more because of inflation.
The Music Room has a nice selection of certified CD players, TMR is an A++ seller of used equipment. Their used CD players would sound much better than any new $200 player you can find.
https://tmraudio.com/search-results-page?q=cd%20players