CDP or DAC?

I have the Cambridge CXC and like it a lot. The remote is not great but you can find a substitute. In your budget range and preferences, I would suggest an Audio Tubadour: https://www.audiomirror.com/product-page/tubadour-iii-se-nonoversampling-tube-dac-2500

I have the MHDT Orchid which I love, at a lower price point.

I picked up a used Moon Audio 260D -- which is both a CDP and transport -- some months back, and couldn't be happier. I love it both when used as a player and when streaming into the DAC. I believe it costs about $3,300 new these days, but can be had for a tad less than half that used, should you get lucky.

I'll say this for sure: I've been playing CDs that I've had forever, and have been enjoying many of them as never before.

(No affiliation with Moon Audio, needless to say -- just a happy listener.)

-- Howard

I use a Mojo Audio EVO DAC (base version) r2r design with a SimAudio 260DT (transport only version) . To my ears very engaging sound.

For what it is worth I had the Cambridge transport in my system prior to getting the Mojo and while it was a pretty good match with the EVO moving up to the SimAudio gave more body and weight to the music . I’d recommend getting a transport that is more commensurate with the DAC of your choosing- YMMV

My Audio Note Dac may be the best purchase I’ve made for my system. I have a DAC2.1x Signature which I had upgraded by an authorised service specialist.
It's paired with a PS Audio PWT transport.

Sublime sonics, smooth, with outstanding realism. 

With your budget, you should be looking at a Level 3 or 4 (used) Dac.

@bobheinatz I run an Audio Note Dac-3 Signature, an older mid-90s version, recently upgraded the output caps and it sounds fantastic. The current crop of AN Dacs sound great and for a lot of us the AN's are an end of game Dac they just sound like music.