What is the most analog sounding CD player under $

The Rega is wonderful. I'm not sure I'd call it "analog" sounding. Very realistic and detailed. If you want to warm it up, there are good tubed models out there such as the Raysonic 128 or the Shanling 3000.

Yoiu might like California Audio Labs. They made fantastic CD players. I still prefer mine to this day. I may take the plunge just to get into the newer formats but the Cal's sound great. They can be had most of the time under $500.

The Meridian is hard to beat for the money and performance. Can be a bit tempermental with reading discs sometimes.

This Resolution Audio CD-50fits your requirements. I don't know the seller but do own one. Yeah we all recommend the item we own bur I've had mine for years now. Can't find better, and I have tried.

Sony XA5400es
Like a modern CD Player/Turntable