Castleofargh, let me know if I understand what you are saying and please correct if I'm wrong: momentary power headroom above the sustained would not occur because the maximum volume is set in relation to a 0 db reference - the maximum volume that can occur in a music file. If the amp can reach this volume without clipping, then it should be able to amplify any signal in any recording at that volume as well as any other amp. This seems quite logical, except that it seems to imply that a given volume always requires the same amount of power, which does not make intuitive sense to me. It seems a driver would be more efficient at certain frequencies than others. I could be wrong about this though.
Disregarding this nitpick, what the other side says is that there are certain sounds which will require power - if only for a split second - much higher than is required to make the music sound loud. They seem to be saying that to go from a low volume to close to max volume very quickly requires extra power. This is not really an argument about volume itself, but rather the physics of the diaphragm. They are saying a signal which goes from low to high will not be properly converted to pistonic motion of the driver unless the signal has a certain level of voltage or amperage.
This isn't all that hard to believe. It reminds me of how distortion sometimes gets lower for headphones as volume gets higher, especially in midrange to treble area. What seems less scientifically supported is the contention that this or something similar happens independent of volume, and is either a result of amps or volts or "cleanliness" of power being less than ideal for a given signal to be expressed in pistonic motion by a given moving mass. I can't give any reason why this wouldn't be the case, but then again I don't know much about this stuff - I can only claim to know more or less the extent of my ignorance.
What really annoys me about this industry is that there must be scientists and industry insiders out there who know exactly how all of this works but are keeping that knowledge to themselves. Or even worse, there is disagreement among experts about how loudspeakers fundamentally work. We need more people studying this stuff and having their work being publicized. I've been trying to answer OP's question for 3 years, but I still don't have really convincing answers for myself.
arf, a lot of things put together here, we'd need a topic for each to get really into how things work in general.
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yes a headphone has various conditions and limitations, that's how you end up with it not sounding like another headphone. the amp outputs the signal and the load(headphone) may let the current flow in some way at one frequency and in some other way at another frequency. the amp can provide that flow of current and it's fine, or it can't and you have power issues. there is no amp looking at the situation and strong-arming the headphone at a given frequency to make it behave against its electrical characteristics. things are how they are. the impedance determines the current flow, the voltage determines the loudness at a given frequency in relation to the sensitivity of the headphone. said sensitivity accounts for everything as it's measured by actually getting the level at the output of the headphone when feeding it with a 1khz signal. so impedance, mechanical forces, acoustic, it's all in the resulting value already.
there is no sending extra power in a secret diplomatic case that Ohm's law isn't aware of.
now realistically, if an amplifier fails to hold up to 5V for a full period of a 40hz signal under a given load, then the amplifier fails at its job in that specific situation where we needed 5V at 40hz into that headphone. the reason is more likely to be about capacitors not holding long enough or something like that I guess. it's a matter of keeping a potential for a longer period, not much that low frequencies require more "force" to push the driver. from that point of view, high frequencies will require more mechanical energy over the same period of time as they "travel" more often.
and yes having specs only for 1khz does not guarantee that the amp will behave properly at 40hz, even less so if the impedance is different from 1khz.
that's why I mentioned that power specs and our little estimation game are only that. we decide about a ballpark value for our needs and check that the amplifier will deliver at 1khz. beyond that, we can only do our best to get amps designed for the use we plan. asking the designer, if it's a small enough brand and the guy replies to mails, is a pretty great option IMO. I have only positive things to say about the honesty of amplifier manufacturers answering my mails or my questions in here for those present. they have a life and probably want to shoot themselves when they start reading my questions, and yet, the vast majority took the time to reply to at least some of my questions, sometimes advising me to pick a cheaper amp in their line, or even not to purchase a new amp at all because what I had was fine for a given headphone/IEM. if they're all like the ones I talked to, great guys!
otherwise, if you wonder about how well your amp behaves with your headphone and if some electrical issues exist, the real answer will be to measure a few stuff. the all approach with guys who sit in a chair, listen for 2 minutes and suddenly think they're the Sherlock Poirot of audio, "it's a lack of power in the kitchen, with the rope!", while fascinating, and correct on occasion with massively obvious stuff, that's not how one should troubleshoot electrical circuits. I feel silly just to state such a captain obvious concept.
it's very fine to rely on our ears only and purchase whatever it is we like most for whatever reason, but then the reasonable thing to do is to avoid making objective claims as objectivity was not involved. again, thanks captain obvious. ^_^
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about the control of the driver's movement, the driver has its very own mechanical and acoustic damping, and with the amp you add electrical damping. the main idea here is that the respective impedance values of the amplifier output and the headphone are what will determine the amount of "control" provided by the electrical signal. when the amplifier has really high impedance the damping will be less. that's really the big main idea. with each damping playing a role on how the driver will move and hopefully, if the driver is good and the damping ratio correct, you won't end up with an horribly underdamped movement. for that specific matter, I'd be more concerned about the impedance output of my amp than about extra power I'm never going to use except mentally to reassure myself.
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you also bring up what could be identified as slew rate and indirectly transient response concerns. so the big idea here is that if we take 2 amplifiers. one able to output 10V with less than 0.2dB attenuation at 1Mhz(or whatever big made up number to carry that idea that it can move fast). then another amp able to do the same thing with a 5V max(we pretend that everything but voltage is virtually giving the same results somehow, although so very unlikely). then it's logical to think that amplifier 1 can go twice as fast
if needed. as the amplitude goes from -10 to +10V even at 1mhz. it "travels" a greater amplitude in the same time or else it wouldn't be able to output signal at the same super high frequency as the 5V amp. so with that idea it's easy to have a mental shortcut with the idea that the more powerful amp is "faster". in reality the slew rate does not have to follow such rule, the amp does what it does and the slew rate should be measured if that's what we care about. the other important point is that we're only dealing with audio frequencies. pretty much any amp is fast enough and then some more for audio signal.
so as is often the case, there is a rational behind those concerns and extrapolations about power. but everything ends up being so oversimplified that the concerns are mostly divorced from practical reality. it's like the great easy idea that balanced amps have more power than single ended amps. it comes from a real idea where the voltage is usually doubled in a balanced output compared to the same circuit single ended. but double what? you can make a single ended amp with whatever voltage output you like. it's not like we can go up to 3V on a single ended output and then we don't have any other option but to go balanced. and yet there is clearly that kind of logic in the air on audio forums. half baked correlations turned universal laws in our subjective reality.
so slew rate is obviously a thing, but if my amp has high power specs then the slew rate will be better and my bass will be tighter or some crap like that, that's 130% logical fallacy.
. but in practice I strongly suggest to disregard that when you estimate your power needs for a headphone, and instead keep using the stuff you've seen posted in the topic so far. 
