It seems that nobody wants to answer your actual question, which was, "I can think of Theta Miles(750-850) and would love it if somebody knew of one, but what else is there?"
I have 2 systems in my listening room, one HT and one 2-channel. The 2-channel rig has a single CD source, so I too have had an ongoing interest in CD-players with built-in volume control that is usable from the remote. Here are the units that I have personally owned. They have all been very nice and all have remote controlled, built-in volume control.
Theta Miles (balanced version-has 2 DAC chips per channel vs. 1 per ch. for the RCA only version) The balanced version is supposedly sonically superior, even when using RCA outs. About $1100 for a nice used balanced version, $800 for un-balanced.
Cary CD-308 The Cary is nice because it has switchable upsampling. Also, it is an excellent transport as it outputs 24/96k via the digital output. About $800 for a nice used one.
Resolution Audio Opus 21 WAY more expensive and out of the price range you listed, but I wanted to say that it is a FINE CD player. If the RA CD-50 or RA CD-55 sound similar, they may be a great value too.
Personally, I have used all of the above with and without pre-amps and in my system/room, the extra cost of having a pre-amp is not justified. And it's not because I can't afford it! I bought the RA Opus 21 because it is A GREAT CD player, regardless of the available volume control. If I thought a more moderate CD player with a passive or active pre-amp was superior, I could have easily bought that setup for less than the cost of the RA Opus 21.
Lastly, just bacause these players have built-in volume control doesn't mean you have to use it. All of these players have been well reviewed over the years and were/are considered great values, even without considering the built-in volume control.