New, a little outside your budget but maybe you can find a used Bluesound Vault 2 then you can stream as well. Is a great streamer with a 2TB drive to rip your CD’s. You can’t throw a CD in and play immediately as it rips the disc first, but after that you can play. Great thing is you can stream and use a streaming service for a small fee and access a library larger than you could imagine and discover new music. The internal DAC is decent, meaning it is pleasant and not harsh but not the most revealing but for the money all around it’s a great buy. From memory it’s as good sound quality wise to the entry Marantz CDP which says a lot.
If you’re hard set on a dedicated CDP, I own and recommend a Cambridge Audio 851C which is now discontinued but if you can find one used it will be within your budget. Built very well, feels and looks like a player that should cost much more than it’s original $1,800 price, later when CDP sales started to drop Cambridge lowered the price to $1,500 I think. Also has digital input which I used to run my Bluesound Vaulit 2 through which was a noticeable improvement. As a player on its own it is a very nice and revealing but not fatiguing but if a recording is horrible it’s not going to fix it. Use quality cables on the 851C and you will be rewarded. I changed out both fuses too which opened things up and sounds a little more organtic.
Not sure they pop up often as those who own the 851C know they’ll never replace it without spending a lot more. I’ll never sell mine. Good luck.