It's interesting you 'miss' something with the high res streams since they typically are of higher quality (24bit/192kHz) than CD (16bit/44.1kHz).
If you hear a difference (very few people can ... do the test over here https://www.npr.org/sections/therecord/2015/06/02/411473508/how-well-can-you-hear-audio-quality ) it could be that you somehow prefer the lower sampling rate of the CD. The solution could be simple and cheap ... switch to the Tidal standard subscription.
It could also be that you hear the differences between the DAC and the electronics of the CD player versus the streamer unit that you use. You could go on the lookout for another streamer / DAC combination that comes closer to the sound that you prefer.