By the looks of your post, you can tell you haven't been around here long.. Please point out to me wherever you've read "it's because one didn't feed them with terawatts of power" See, what you're doing is feeding an age old myth. This is how old myths stay alive.
I along with countless others have tried to kill this myth thousands of times, but it just won't die. Any headphone amp, speaker amp, portable amp, whatever amp - can drive the HE-6.. OK. However, it will not sound good out of most of those amps. What we are trying to do here is get the most out the headphone, trying to reach it's absolute highest potential. The only way we've found to get there or very close is by using speaker amps.
Now, I've owned a 10watt into 8 ohm speaker amp that I used for over a year with my HE-6. It's still one of the best amps I've heard with the HE-6. That's right only 10watts into 8 ohms. Guess what that does to your "it's because one didn't feed them with terawatts of power" statement? That's right it just killed that myth.
What you need is a capable amp. A capable amp can come in all sizes, shapes and wattage output. Still - it has to be capable. Capable of what you ask? Capable of sustaining consistent peaks in some music passages without running out of gas. Capable of taming the much talked about harsh highs. Capable of taking the bass of the HE-6 by the balls and controlling the hell out of it. Capable of allowing the mids of the HE-6 to sing baby sing.
There's a hell of a lot of capable amps out there. Just not many headphone amp are capable. Your amp best the have a hell of a power supply with some hefty transformers. Most headphones amps of yesterday didn't need to have this. It's now today that headphone amps are putting out more power, but still not sure if their power supplies can keep up.
The EF-6... The EF-6 has more than enough power to drive the HE-6. However, to many the synergy is just not there. It's still a tad bit on the bright side. Still does not give the treble it's just do. I had the EF-6 in house on loan for about 6 months. I put it through all kinds of test, side by side with the GS-X mk2, and the lowly 10watt into 8ohms amp. Guess what.. That lowly 10watt into 8ohm amp kicked it's ass. The EF-6 and the GS-X mk2 was going toe to toe with the GS-X mk2 winning by a hair because of out it tamed the highs. However, the GS-X mk2 would sometimes run out of gas on extreme busy passages.