I think the big deal here is how when somebody posts that the HD650 needs a special amp with... big power number, OTL, balanced output or whatever other stuff he considers to be the legitimate cause of better sound for usually anecdotal reasons, he's in fact just talking about what he prefers using. which is often not even remotely related to concepts of fidelity or objectively improving electrical and mechanical behavior for the hd650. IMO, declaring that the hd650
needs whatever special kind of amplification(beyond basic power requirements for a sustained given SPL output), is the same as declaring that pasta
needs tomato sauce. it's very fine to have that personal preference, and chances are that many do share that taste, but it's still only taste. there a line to draw between that personal preference, and going to declare a universal pasta fact on a forum. otherwise don't act surprised when the carbonara squad invades the topic to defend their own views. then there is always going to be the group thinking that some butter and a little salt is the best. and sooner or later we'll have a post from that one guy who likes to put his pasta with fries into a sandwich and doesn't get why nobody else does it.
IMO, the moment someones starts discussing taste in a way that makes it look like a fact about the headphone itself, any sort of argument is going to end with him being wrong for doing so in the first place.
I remember some brave member once posting about how he/she enjoyed the hd650 straight out of the computer's integrated soundcard at night because it gave a more ... "boomy?" low end or something like that(very likely related at least in part to the impedance output as you mentioned). that person's hd650 didn't
need anything else at that moment.
with that in mind, I'm tempted to agree with President
@Eisenhower's main point. but of course certain amp designs can and will have an impact on the sound of the HD650. some will like it, some won't, some will think it's a big deal, and some won't. on that you're absolutely right too. it's just not a need for the headphone. and if some special amp does lead to certain objective improvements, I believe we deserve to see it demonstrated with measurements and not just proclaimed out of nowhere.