The sound quality attained through a high end headphone system has similar requirement to that of a primary system. Although the demand on your amp is not as high. The demands on a streamer and DAC certainly are. Headphones are very sensitive and put the drivers right next to your ears… so You are going to hear th SQ. Especially with really high-quality drivers like the HD 800s.
You can see my headphone system under my user ID. It produces sound quality at the level of my main system. I have found with this system the Streamer and DAC are critical. Over the fifteen years that it took me upgrade piece by piece until it really became a serious system with really good and magical sound was to have a really good DAC… I tried all sorts including ones like Dragonfly, a small two box “high end “ ESS solutions from HeadPhone… and a number of others. It wasn’t until I got a real high end DAC (Ayre) that the sound really improved to become high end. My experience with all inexpensive DACs has been a real disappointment. My recommendation is to get a real DAC (mine is half sized)… something like a Schiit Yggdrasil or better. Then follow with a high quality streamer.
The sound from my Sennheiser HD800’s and Focal Utopia is truly wonderful and without fault. Getting there was a long journey, and cost money… but the good thing is it was still only around 10% of my main system.