What you describe is not a characteristic of the good Class D amps I've owned. I'm currently using PS Audio M1200 monoblocks and they are fantastic. Of course those are 3 times the cost of your integrated amp.
Cheap Class D devices may sound cheap. You've compared to a number of other devices and they all sounded better. That's not an inherent issue in Class D, it's an issue with that specific device. Is it too late to return it or replace it to see if the replacement unit is any better? If not, return or dump it and move on.
I had a similar Class D integrated from another manufacturer sometime ago and it lacked body and clarity. You may have bought a lemon.