Well i guess i just comes down to personal preference after all. I do like my k702 in both tonality and detail, it's less of a "less bass=more detail", but rather "less bass = me more like it". I'm not very sure about K612, as i don't own them currently, but comparing K702 to HD600 both of which i have at hand right now, i would choose akg over senny, and i would choose them both far over K371 which i keep mostly because it's sometimes fun to stirr things up a bit.

Anyway, as i said, i did try using my K702 with both my amp and my phone. I admit, the difference is a lot less than i expected, i didn't notice much difference in neither bass nor mids. I did, however, notice difference in treble, from my phone (Redmi note 8) the treble is sharper. The problem with my methodology is that i am using different DACs while changing AMPs. I think, that the extra sharpness might be because of lost detail, as with K5 pro the treble is effortless, while with my phone it's... Clipping? Might be, i'm not shure, but it definitely results in an overall sharper sound.

So you were right, it does seem that amplification of headphones is all volume. I really want to try now my KA1, to finalise this, but it's broken, so better luck next time. Probably my experience after buying K5 was due to a better DAC in it, which resulted in less harshness.