Short and semi-wrong simplification: a small amp, like a phone-amp, can produce effect peaks that are sort of technically sufficient to blast your ears off even on a pretty high impedance driver (like on the hd600). But sustaining a more complex sound-picture, and still creating the dynamics (i.e., the differences in volume that are, presumably, in the recording of what isn't an electronic metronome) needs a longer and higher effect threshold (that on normal listening volumes might not be very high, but still significantly higher than a phone-amp).

So.. when you test the physics of it, you could get results on a very light amplifier - and that would even correspond to your listening experience - where the amplification required to produce an accurate enough tone for a short amount of time is extremely light. I've done some testing that is similar to what produces the effect-curves and frequency responses you see on some of these tests - and for short enough intervals on reasonable volumes, a massive effect threshold isn't necessary. At all. Like a 4W laptop amplifier competes with a Chi-fi star from Marantz that happily drives two large front-speakers on 2x100W until the glasses ring in the cupboard.

But if something more complex happens in the music (and the source you're playing back has a higher density in the actual recording - and not that it has been "upscaled" to 320kpbs mp3 or something like that - so that you're even going to be able to hear anything here other than the boom on the lower tones, and various other things that are just not produced in the actual speaker-driver) -- then the tones/music will change once you have a sufficient effect threshold to draw on.

It still doesn't need to be very high, which is why a decent enough carryable amp is enough (and why Chi-fi is a thing - the level required for more than decent enough playback is not actually that high. And the design-need to put in incredibly high ohm speakers also isn't really there, because people don't have power-grid noise or things like that that really cause noise - we're not at that level in recording or monitor sensitivity on your typical home-product that plays back, at the very most, 320kbps compressed audio, sent through any amount of noise-filters on the way). The idea that a cd-recording, that was compressed down to a cd-format in the first place, that then was washed through the cd-player's horrible variable codec, that then produces a signal that is washed with a noise-filter before it's amplified, and then also reduced slightly again through the amp -- needs a million Watt amplifier from Klipsch or Hegel to sound good on a pair of normal-sized speakers in a living-room is beyond ludicrous.

But basically, on the first step, you're looking for an effect output that can make the headphones work consistently. If you have to turn the volume/effect very high up to get much of anything, you can hear the effect very easily when you play something else than a test-tone, in that the difference between loud and quiet is basically not there, or that you're missing reproduction of some frequencies when there are other things happening as well. I.e., you get the test-tone, but while doing two, there's a variation in the output.

The second step is getting something that has a high enough threshold so that in complex parts of the recording (some people notice their blip-boop music not having a great enough bass-crunch while the keyboard is tapping an a-tonal horror that would make Schoenberg flutter in delight) will be the same when you add another element to it. On classical recordings you could have the same effect because of noise around where the recording was made, for example. So that - if the recording wasn't leveled through a press and then cut out in an a4-shaped cutout - that noise from the chairs and various echo-bits and resonance would actually sound fantastic on a good setup (as much as the recording equipment allowed). But when played back on a kitchen-player (the gold standard for "modern" digital formats), the recording would sound inconsistent, unpleasant to listen to, and dry and empty.

That effect-threshold for reproducing something with dynamics on a 300ohm headset is not very high. But it is higher than a phone-amp's, which is typically reaching peaks of the effect the power-supply can handle or is designed to handle at around 20ohm. The "gold standard" for mobile phone headsets, as with bluetooth headsets, is basically then to produce headphones that are in the 16ohm range.

That doesn't mean they suck, it's just that higher impedance speakers would sound like trash on that system (in certain situations). To make things worse, though, some of these large headsets also work, and are designed to work on lower effect. So you don't get muted or garbled tones on an under-amplified system (which again brings with it an endless amount of design-for-most-common-usage drawbacks). It's just that you're not getting out what the headset can produce.

Which admittedly very often isn't an actually high level. For example, if you can have a sustained effect without drops through, say, 5 seconds, on a very low effect amp (such as a phone-amp - they do exist), you can actually get away with that on low volumes on fairly high ohm speakers. Because what you were looking for was not to produce peak effect on all frequencies, in some astonishingly modern music recording that is unfit for human consumption -- but a lot less than that.

Or put in a different way, the difference between "hi-fi" and "crap" was actually the amplifier's ability to output the desired, still very low, effect for 1 second, instead of having a drop-off after 0,3 seconds. And that then retained all the dynamics at a humanly listenable volume, produced all the detail, etc., even though the amp is not going to melt your ears (at least literally). Note that you also have these effect-drop offs on more expensive amplifiers, just in a different range.