I think this all depends on what your preference of HEK to be a bit forward or not. Headphones generally have slight differences in how it's driven, like for example in the case of HEK, one may be perfectly happy driven straight out of a Mojo, but with something that drives harder, they may find enough differences to warrant the amp's performance.
There's some stuff that doesn't make a whole lot of sense in general, especially regarding HE-6 being popular with speaker taps. HE-6 is quite inefficient, so there are some partial reason for that, but how much volume are people using even from speaker taps(although I guess the wattage varies for different speaker taps)? Balanced outputs and also higher gain typically sounded more forward and punchier in my past experiences, and this may just be due to greater gain overall. Like for example, 30% of the volume dial on my amp on low gain SE maybe 10% if on balanced at high gain. But subjectively, balanced just sounds like the ambient sounds come out more for live recordings, but also the sounds gets more forward. This is something I've notice for live recordings when comparing amps as well.