i was very impressed with the vincent s1.1, which is a hybrid tube player--very natural sounding. they also make tubed players (probably available used at your 1k target) which are well worth checking out.
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.
You got a lot of good suggestions but I question the premise of your original comparison question. I've owned a few different turntables and CD players over the years and I've had CD players that sounded more "analog", or in general much better than some of my turntable/cartridge combinations. Sorry, don't mean to derail this conversation but I'd like to hear someone explain what "analog sound" sounds like.
I found it! In an antique shop covered with rust and gunk. The transport had to be prodded with an ice pick to get the tray to open. The CDs skip, distort, and the music can barely be heard underneath the cacophony of debris. Just like vinyl!
What do you mean by isolation - physically or electrically? To the OP, I'm also in the same situation as you and have decided to look for a better DAC than a "better" CDP. I think that approach provides a better longer term path for future growth.