I'm by no means an expert, but thought I'd chime in before I embark on an attempt to catch up with uni work desperately. One iffy bit, I really wish you paragraphed your text, but that's another thing altogether. I will not be commenting on your own equipment as I don't have much experience with them and I've only briefly listened to the dt 770pros and I didn't like them much, but for the deal you got, I'm sure it's good value
.
Anyhow, from my knowledge, there are different links to the chain. There is the source material (lossy/lossless) then the source (dac) and then the amp, before the headphones/IEMs. i.e. source material - source - amp - headphones.
A lot of all this is subjective and I'll try to be as competent and coherent in laying this out and hopefully, in an understandable manner. I'll be breaking it up into the chain mentioned above so I'll only talk about that specific unit and then from that unit as to how it affects the units before itself in the chain. Also, please note that I will be trying to go about it in simple terms and hopefully, not convoluted and I will be missing out on more in depth material, so this is really just to me, at least, basic theory.
Source Material
Firstly, I'll start with the source material. What's frequently said on the boards is that the sound will only be as good as your source material and I couldn't agree more. Therefore, if you feed it something that has rather absurd compression rates, you will more than likely find yourself to be listening to rubbish. However, there are several threads going around where 128kbps was tested against lossless and many, including myself had trouble discerning which was which and at the end of it all, the consensus was that 128kbps encoding has gone on in leaps and bounds compared to the days of yesterday and some people are still unaware of this.
Source
This is where the DAC comes in. What the DAC does is it attempts to resolve the sound. Some DACs up-sample the bit-rates, some clean up sound, some just portray the music as is. Some DACs may colour the sound and making it seem warmer or more "musical" to some people. More often than not, the DAC deals with instrument separation and usually enhances the sound with slight manipulation (may not be the best word) of the highs and lows, etc.
Amplifier
There are different kind of amps, SS and tube and it is often seems to be that the tube amps are warmer in sound signature. But essentially, the job of the amp is to take the signal from the source and bring it up to speed with your headphones and in the case of a SS amp, opamp rolling is a frequent occurrence. What opamp rolling does is to change the sound signature slightly. Some tighten the bass, some loosen it and the same goes for the treble etc. But these effects are subtle on some headphones and less obvious on others. E.g. If I had a bass heavy headphone, tightening and taking away sub-bass would be a difference that can be heard, but where you have one that is bass light and with bass roll off, then you probably won't hear much difference seeing that the sub-bass was probably non-existent in the first place.
The other job of the amplifier is to power your headphones. With enough "juice", the amplifier is able to drive your selected headphones to optimum power and optimum effect. Not all headphones require an amplifier to sound their best. A power hungry headphone might struggle with a basic setup and what happens is that the headphone just sounds lethargic and thin when compared to being powered by a amplifier capable of meeting its needs. The thinness will not be apparent until you compare it to an amp which matches the headphones. But when this does happen, often the headphone will start to open up, even in soundstaging and with better instrument separation, but in my opinion, it is only because it is finally being fed with enough "juice" to reproduce the effects of the DAC on the source material. Yes, some degree of added separation may be from the amplifier itself, but this is where there is some debate as to which is more important, the DAC or the amplifier and that is really dependent on headphones to some degree.
Headphones
These are the last in the chain and to be honest, I'm very much in agreement with you that with decent source material, these are the difference makers in the sound. Again, it boils down to the headphones only producing what the source material has fed the other links in the chain, but the headphones are the ones that bring the character and its own sound signature, e.g. being bass heavy, light, treble brightness, etc and other flavours.
More often than not, the headphones without the DAC and amplifier will usually sound similar to when it is paired, but the major differences when pairing it right are to some extent, due to the fact that many are attuned to good sound. But at the same time, without a good amp, assuming your headphones require one, they will sound less full-bodied and less energetic when they are then fed a amp which synergises well with it. That is what brings the headphones to life.
The differences from the DAC and amplifier are more subtle because at the end of the day, the headphones already have their own sound flavour and that will only change so much from a DAC and amplifier unit. In a way, the DAC and amplifier work like an EQ, not totally, but in some ways in that it manipulates the sound a little. The drivers in the headphones are the limit of its capabilities. In a bass light headphone, no amount of increasing the bass via EQ or other equipment is suddenly going to make the sub-bass come to life to sound anything like a headphone that already has sub-bass.
Conclusion
The headphones are the life of the whole chain. If you don't like the headphones without the DAC and amplifier in the chain, it is still probably unlikely that you will like it when it has the right equipment because the sound signature and flavour of the headphones are already there for you to hear, it doesn't change drastically. There are exceptions of course.
Alright. I think that is all from me. I'm open to correction and for people to question, etc. Please do remember that this is my own experience, I'm trying to keep it 'short' and simple and I'm by no means an expert in this field. I only enjoy my music and I think that is the most important in the hobby and probably the best advice I can give.