to get a proper response, I believe we would need to measure the output using a given set up, or at least know very well what design is used in the amp and have plenty of reliable specs.
often, changing the gain doesn't really show much difference unless you're borderline of clipping/distortions by reaching the limits for nominal behavior of some components. and if you've bought a fitting amp for a headphone and DAC, then you should have a lot of amplitude to fool around before reaching those types of situations.
on some amps the gain settings exist to match the input voltage of the DAC(that usually wanders between 1 and 3.5Vrms depending on DAC models). so that's a different matter but input gain is a thing.
the type of volume control could also play a role, some will have channel imbalance at lower volume setting, so reducing the gain lets you turn the knob higher and often get better balance for that kind of knobs.
about the added noise from higher gain, you're right on principle. just boosting the voltage more to then just force the volume knob to attenuate a lot more too, that doesn't seem to make much sense and could potentially increase all kinds of noises. but again that could depend on the design, the O2 amp is doing just that, keeping stuff high from the get go with the aim of keeping as much dynamic as possible all the way through. and even if it can seem counter intuitive, the result is a surprisingly low noise floor. but of course by keeping everything close to the max, if the DAC has a higher voltage than expected, then you go straight into horrible distortions and a matching input becomes more important than it will be on typical amp designs. so again, different designs can have different impacts and different needs.
and those are only a few ideas of what could matter on the top of my head, so it's hard to make an absolute claim on this subject.
still the general consensus for voltage amp, everything else considered stable and properly paired, is to use the lower gain and increase the volume control. then when you reach the maxed out loudness and still want it louder, you switch to a higher gain setting and find out if it's ok. that is the most common logic behind an output gain implementation.
the sufficiently powered part you talk about is both true and total BS at the same time. for typical voltage amp, the gain is a voltage gain and voltage is directly related to loudness. so at a given loudness in a proper system, the load(headphone's impedance) and loudness(voltage) are the limiting factors in the circuit and will determine the power applied for a given amp. it means that the louder you will need to go to enjoy your music with a sound system, the more power you will need. that much is intuitive and everybody gets it.
but it also means that for a specific loudness, the power is fixed into a given headphone. so if you can go loud enough on low gain, then high gain is not going to send more power into the headphone once the volume is set at the same loudness. and that part is usually misunderstood and leads to people talking nonsense about more power. often even mistaking the specs of maximum available power for the actual power used, and other funny stuff like that. a little like thinking that a 800W PSU inside a computer is going to send 800W to the DVD player. that's just nonsense.
of course voltage isn't everything and the amp could fail to deliver enough current, but will increasing the voltage gain, provide more current? should we call it current gain then? or power gain? ^_^
again different designs might lead to some changes in current, impedance output, etc. so you'd need to know the specifics of a given design or simply measure the output of the amp into a given load. but it's plain wrong to just expect the voltage gain switch to be some magical "turbo" button that boosts everything into betterness. that's just ignorance talking most of the time.
last but not least on the web:
you have probably heard about the psycho acoustic effect of loudness, the famous "louder is better". this has nothing to do with the gear and all to do with the listener. when listening to music, louder(up to a point) feels superior, better bass, better soundstage, clearer details etc. that's how it feels. the average Joe uses his amp, presses the gain switch back and forth and subjectively experiences all those merits of louder sound with higher gain. I'd say most people simply fall for that effect and make up a fallacy involving power as to why it feels better.
to be able to tell anything about the sound without falling victim to the louder is better, we'd need a blind test and matched loudness. something that pretty much no audiophile giving his opinion has ever done. so from an objective point of view, you shouldn't care too much about those feedbacks, and when it comes to power, better trust good old objective electricity laws.
