Yes, this would have been at a pretty high level as that section of music is at a high level and, again, the only headphones that handled it was the MDR-1R. The level was high enough I could feel, not just hear, the vibration of the very low lows -- somewhat like the "feeling" you get from very low LFE from a subwoofer.
This is what happens when headphones have high harmonic distortion in the bass (and if you think about what harmonic distortion is, it makes a lot of sense). It's a poor and often deceiving substitute for that feeling you get from an actual subwoofer. It technically isn't
accurate, but, as I mentioned earlier, many find it tasteful. Knowing what you find tasteful from an objective/technical perspective is a good thing. Nothing wrong with liking what you like, so long as you know what it is you're actually liking (most of us don't really have a good grasp on what truly neutral/natural sound is, and we all have slightly different inherent tastes). I have purposely sought out audio gear that is poor by many objective standards but better suited my needs and tastes. (And there's enough consensus in MDR-1R measurements to assume the measurements for it are probably fairly accurate.)
I totally forgot the mention the NAD VISO HP50. At $300, they're getting some really good reviews and might have suited your tastes if you like the MDR-1R. I already mentioned a couple options that are technically better within your budget (Mad Dogs probably being the standout).
You are right, though, that not many headphones measure "flat," but there are some that come pretty darn close (HD600 or, on the darker side, HD650, Paradox modded T50RP, the new Focal Spirit Pro and Classic, etc.). Even then, by "flat," which compensation curve are you using? (Harman has been doing a lot of interesting research lately on what compensation curve and overall sound signature people prefer from headphones vs. speakers in a room.) You also have to understand what software and hardware setups are using for anyone doing headphone measurements. For example, "flat" or "neutral" headphones measured on Innerfidelity will actually often have a slightly downward slope to the overall frequency response (not always, but often). The measurements from that other audio forum are based on a different hardware setup and use different compensation curves. So, it's messy reading measurements, especially from different sites, but SOME valuable and accurate information can be understood.
The MDR-1R's frequency response itself isn't bad. The only real negative is the mid-bass hump, but it's not crazy enough to be a big deal. However, the harmonic distortion is quite high and disappointing for a headphone of that price. THAT is where I'm primarily "concerned," not with the frequency response. Still, you can literally find a headphone that performs similarly but with overall lower harmonic distortion and flatter bass for...$30 (Tascam TH-02, though the stock pads aren't great). And it gets noticeably better over its own stock performance with mods (MDR-1R responds less to damping mods, unfortunately).
As for the buzzing? I have no idea. Given the MDR-1R's performance, it's more likely it's masking something than revealing things as they actually are (I have heard "buzzing," often clipping, manifest itself on nicer headphones that poorer headphones could not resolve properly). But, that's not necessarily the case here. You also have to consider the condition of demo headphones, the source material you are using (music files and playback hardware...were you using good files and a good DAC with a dedicated amp?), volume settings, etc. A phone + portable amp really isn't a great setup for higher-end headphones. I'm also not sure if you were using EQ during your tests, but that can really mess stuff up. I don't find the HTC One to be particularly stellar for audio purposes. Are you sure you had the Beats EQ stuff turned off as well? Does your portable amp use USB or is it from the headphone out (if the latter, that's going to hinder your performance)? The headphone out on the HTC One is a bit harsh and grainy to my ears. Bass response isn't great. Decent enough for sensitive, basic headphones and very casual listening.
To comment on some headphones you tested:
HD650 - Requires a good DAC and a good amp and can be picky in this regard. NOT a good portable headphone. Requires a quiet environment for good performance due to being open. If you don't have that, don't bother. I don't recommend this as a general-purpose headphone...it really is best with good gear and a good environment. Without a good chain behind it, you can expect poor performance, especially with bass. Even then, the HD650 is tuned a bit dark and a bit sluggish with bass at times, from what I have read. Still, a fairly neutral, well-behaved, relaxed headphone if you feed it properly.
DT770 - No experience with any Beyer headphone, but these have an elevate bass response and harsh, elevated treble based on most measurements. I see these recommended less and less as time goes on. *shrug*
K701 - I tried the K702 anniversary edition...basically same headphone with minor improvements, namely in the bass. I did not like these...too much of an emphasis around 2KHz that killed my ears. Nothing really wrong other than that, but also nothing really great. They do really poorly at loud volumes (way too much additional harmonic distortion). Very picky with the DAC and amp chain.
EDIT: Did you end up getting the MK1 or MK2 of the MDR-1R? I've only seen one measurement for the MK2 (from a website that has some iffy results at times), but if you got the MK2, clearly you could be hearing something different that what these measurements indicate (I believe most are for the MK1).
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