OK I am going to suggest you try a sony playstation. I know a lot of people here are probably sick of me recommending this as a player I don't care, for the $20 you will have to pay to try one what have you got to lose. Chances are you might have one lying around you could try right away. The reason I suggest this player to you is because you must like valve sound and the playstation has a sound more like vinyl than many other players and there are a few companies using the playstation coupled to a valve output stage and it reatail for a couplke thousand dollars I believe. Also you say you can't really hear much difference between the cheaper players, well this one is very cheap and I think you might hear a difference.
There is another thread on this subject in the DIY forum, but if you want to try it look for a very early model playstation 1. You want model SPCH1001 or 1002 which has RCA sockets on the back. This player sounds the best of the PS1s IMO but some of them droip out occasionaly on some CDs I think the laser bias is manual, the SPCH-5502, 5002 rate highly also and don't skip generally they have auto laser bias. Last choice would be a 7052 model, I wouldn't bother with any of the later models, they made them cheaper and changed the design sigbificantly as time went on.
You can use a PS2 DVD remote control unit with the PS1, it has a reciever that plugs into the controller port. You can just use the controller to try one out though, no need to plug it into the TV, just press the start button to play and the buttons on the top of controller to skip tracks.
Please report back if you try one, also if you do try one I would like to hear your opinion compared to a higher end CDP if you decide to get one. Also do you DIY? there is lots of DIY DACS non OS DACS, reclocking tricks etc you can do to commercial units or build from scratch for relatively low cost and a soldering iron.
good luck.