What makes a headphone objectively good sounding?

[AudioMastermind]

Assuming tonality is more or less completely variable, given you have enough amplification power and low enough distortion transducers, what defines the next dimension of sound quality?
1. Magnitude frequency response is consistently the number one predictor of sound quality across all metrics. If ultimate sound quality is your number one goal, parametric equalization is a cheat code to achieve it.
2. The surface area of the headphone driver interacting with your ear tremendously affects perceived soundstage size. It is not only perfectly logical that many natural anatomical head-related transfer function (HRTF) features account for themselves but that there will be an increase in tactility, which is provable by measuring the frequency response at points outside the ear (something you cannot do with an In-Ear Monitor).
3. The leakage tolerance defines how the headphone's low-frequency response changes in response to changes in the seal. A headphone with a high leakage tolerance achieves good bass response across a variety of head shapes. The HiFiMAN Susvara is a great example of a headphone with a super high leakage tolerance.
4. A headphone with low magnitude frequency response variability across reseating in the midrange and treble frequencies and high response smoothness is generally the best platform headphone to use if you are willing to use DSP correction, as they require little computational power to tonally perfect. The Audeze LCD-5 and Moondrop Cosmo are probably the best headphones in this aspect, aside from the Sennheiser HE-1, which arguably doesn't need DSP correction.

 

[Killcomic]

What makes a headphone objectively good sounding?
Nothing. It's all subjective.

 

[castleofargh]

What makes a headphone objectively good sounding?
Nothing. It's all subjective.

Just because the judgement is subjective, doesn't mean it won't correlate with at least some objective factors.

Op's propositions are:
1. FR is the main predictor of preference. Yes it is. It gets complicated a little because not everybody wants/need the same FR, and because 2 different headphones with the same FR on a dummy head probably won't have the same FR at your ears. So no magic bullet for that one. But a FR you like will always be relevant to how much you enjoy the headphone. That aspect isn't controversial.
2. Is correct with OP's premise of low distortion. A bigger driver tends to help get a better, more stable response. Consistency is always desirable, isn't it what hifi implies from the start? It might be true that having a bigger skin area receiving the sound could be a factor(IDK if that has been tested, but the hypothesis has been around). It's at least accepted that the ideal driver is big, light and super stiff. That's every designer's dream. The issue being that IRL we always have to compromise somewhere, if only for durability. There is a video from Sennheiser where for their research on dynamic drivers, they settled a certain diameter because of physical constraints.
But even ignoring everything else, if you use a headphone with a tiny driver, any small change in placement will affect the sound more because the driver will align differently with the ear. A bigger driver managing a bigger wave front would mitigate those changes from placement. And that brings us to OP's 4. and my initial comment on stability.
3. again is about stability, if from one placement to the next or wearing glasses makes you lose like 6dB at 50Hz and more below that, your favorite sound of yesterday won't be nearly as good today. We can always move the headphone around until it's fine, but it's a bother and there is no guaranty that everything will stay in place once you start moving your head or lean back to listen. And yes, that is one of the results found by Harman. But it's also just common sense and maybe if you have used a variety of headphones or pads, you already had observed as much.
4. again is about aiming for stability, so that once you have found the sound you like, you can keep consistently getting it.


I would put comfort at the top because I can EQ some of the FR issues, but I rarely can turn something uncomfortable into something that is.

 

[bigshot]

The main features that are important are frequency response, comfort and compatibility. All those things are important. Everything else is secondary. If you can find cans that sound great, you can wear them for a long time without getting tired of wearing them, and you can plug them straight into your sources without needing a separate amp, you’re home free. I am there, so I don’t need to keep looking.

 

[Whazzzup]

Everything leading up to the hp, then the hp preferences.