I'd say its usually between 85% -- 95% headphone, and 13% -- 3% amp, and about 2% DAC. I heard someone once who said 98% driver/headphone, and 2% amp. It's how you want the sound shaded, or what gaps you want filled in. DACs are even less a part of the sound, maybe just a bit above cables (no pun intended)
I tend to think the headphone is about 95% of the equation on average, and 5% is "everything else, mostly the amp" but its the whole system working in synergy with good files that really makes a great listening experience. I'd rather have a thousand quality, well-recorded FLACS I enjoy with an adequate amp, than a really good amp and a bunch of 128kbps files that were recorded in someone's garage on their jury-rigged potato-clock microphone. But that's my two cents.
DACs are an enormous part of the sound.
Headphones are just speakers. They don't actually make the sound on their own, they're just there to transmit the bits of data being fed to them after it's been translated into analog. It doesn't matter how amazing your headphones are if the music being fed to them is bad or the signal is weak. You can throw FLAC files at a crappy source without enough output power all day long and it's never gonna sound any better than those 128s.
Remember, everything you listen to through your headphones comes via a DAC and an amplifier. Everything. The question is whether the INTERNAL hardware on whatever source you're using is adequate. Plugging your headphones into a Mac Pro is going to yield a wildly different experience than plugging them into a $200 Chromebook. Suggesting that DACs and amps are unimportant is foolish. It'd be like making a car audio system in your 1987 Civic and acting like you can still keep the stock dash unit an expect everything to sound great.