These are the main options -
1. buy a brand new CD player. These are now an esoteric hifi item which sell for many hundreds if not thousands of dollars.
2. buy a second hand CD player. These either seemed to be newish and still quite pricey, or very cheap but anything up to 25 years old, so probably no more reliable than what I already have.
3. buy a DVD player. They all play CDs and a decent brand (eg Sony) can be had for $80. The problem is that the audio quality on them is poor.
Then I came across a NAD T531 second hand. This is a CD / DVD player from 2001. The difference is that it has high quality D-A converters for the audio output. Problem solved.