Natural or closer to natural w.r.t. frequency response is a measurable entity. You can make flat measuring speakers, and in an anechoic room you can hear how those sound. When you put such speakers in a room, you hear the power response, a combi of the direct sound and reflections. The response is not flat. Floyd Toole writes about an overall preferred curve that is considered to be "overall satisfying", with a bump in bass, and going down like 2dB from 200Hz onwards.
Free field, diffuse field, and all sort of variants of Harman curves are an attempt to come to a "power response" definition of a "natural" headphone response, heavily relying on personal properties like ear canals, ears, head shape and shoulders resulting in tens of dBs of difference in a frequency range.
I develop loudspeaker for decades, with a flat on-axis response (anechoic room measured) and a balanced power response. Difference of 0,5dB in treble are super obvious, and with some recordings you tend to go up in treble, with others down to get to something I experience as natural (tweeter curves start dancing up and down from 3kHz, so what is the proper average sound pressure to claim it is "flat"?).
I go to classical concerts at least once per month, realising that every voice, cello, violin or full orchestra etc. sounds very different. Even the way a pianist plays the same Steinway piano in that concert hall (Uchida, Pires, Solokov, Yuja Wang, Giltburg, etc. etc.) makes the piano sound super different (intonation, force in the lower registers, use of the pedals). Sitting at another place in the hall as well. So, tell me, how should a piano sound like, if you were not at the specific place at the recording, and understand the microphone placement and mastering choices?
It is not so black and white. Large variations and distortions are definitely easy to recognise, as they put an explicit signature everywhere. But subtle tonal variances, that may make-or-break the subtleties of a musical performance, will largely differ per instance.
Now on top, there are many other thing than frequency response; e.g. impulse and/or phase response, resonances, noise floor, linearity over volume/frequency, intermodulation, driver breakups, left-right coherence, spaciousness, etc. etc. that add to the concept of naturalness. Which effect is more dominant, and is this true for all sorts of musical performances?