Per a quick check (not experience) these seem to have neutral to bright tone profiles. The CMA-400i uses the very common AKM4490 DAC chip, while the Klipsch uses an ESS 9018 chip. The ODAC is an older design (circa 2012) with a 2015 revision – ESS 9023 chip.
The Denon AVR-687 receiver appears to have been released in 2007, and the documentation focuses on its home theater functions. I didn’t find any details on the headphone technology. The odds are that it would be more mid-range focused versus the headphone systems you heard (i.e., less bass and less treble content or clarity). Many people do indeed find technically accurate amplifiers to be unpleasantly analytical, clinical, or harsh – this often pushes them toward tube amps.
Were they all fed by the same quality source music? If any of the tests were based on compressed sources (e.g., MP3, Apple, etc.) they’d generally sound worse.
This partly follows from personal hearing, partly from experience, and partly from hobby interest. Many (myself included) started off with similar low cost equipment and then upgraded to $2K or $20K headphone systems. Again, if you use basic sources or have a limited amp or DAC then you may not hear the differences. Learning follows from exposure, trial, and error.
My hunch is that you’d prefer a warmer, softer, smoother tube amp too. The amp threads have recently covered both tube and tube hybrid amps.
Specific setups will be warm, neutral, or bright. Some will be “V” shaped to emphasize treble and bass while others will be flat or mid-centric (e.g., likely your AV receiver). Tube amps will generally be smoother, warmer, and easier on the ears (excluding many of the $50 tube amps on Amazon or Ebay).