Another question. With all three of the headphones I've mentioned, K701, T1 and HD600 - the first thing I've noticed when they aren't amplified is how thin they sound comparative to hearing them from a decent amp. If the 100hz frequency is actually easier to achieve - then why wouldn't they sound warm and overly thumpy rather than thin and weak (like the bass has gone AWOL)?
I'm assuming you've got your impressions listening at fairly similar levels
I think there's no universal answer for your question.
I can only share some views and personal experiences.
Some low grade amplifiers have significant roll-off in the lower frequencies.
Some smartphones cut-off at 60Hz by default (not sure why).
I've tried warm iPods, and slightly bright iPods too.
Not sure what amplifiers you've used when comparing, but some (quality) high end amps are indeed slightly on the warm side of things (some Violectrics for example)
I've tried HD600 and DT880/250 plugged directly into a low grade Samsung Galaxy Young (S5367).
Far from thin both sounded full and darkish/veiled/not clear to my ears, but bass wasn't lacking, it was more on the bloomy side specially with HD600.
I've also tried DT990/250 plugged directly into a Lenovo laptop with similar results, less clarity / more diffuse / softer presentation but bass was still there, although not as detailed.
Most of these low grade devices offered less contrast than a quality DAC/Amp, and contrast naturally bring up the perceived bass impact.
Going from Fiio E10 to Schiit Magni some time ago, the weaker Fiio was slightly warmer to my ears.