The two balanced analog inputs on the 812 is off course kept purely analog (and balanced) to both the headphone amp and the balanced output from the pre amp. It's not like the Unity Atom headphone edition where the analog input is digitized and then converted back again (a serious corner to cut in my opinion).
I have been using my 812 purely as a headphone amp for more than 7 months now, and mostly with the analog inputs (CD's from a BCD-3 and vinyl from a Whest phono stage), and I'm only using the built in DAC with streaming from Roon for inspiration and checking out new music. The headphone amp in the 812 is very very good, and to me that alone makes the 812 worth every penny, but the DAC is also very good and I could easily live with only the built in DAC, also for CD's from a proper transport.
I have not tried to connect by wi-fi (with the wi-fi dongle), but in every other context regarding music reproduction, wi-fi has always been inferior to a hard wired ethernet connection, and I agree with hifixman that the 812 deserves a proper network switch with a decent ethernet cable.
I have never considered a wi-fi connection for streaming, so I don't have any good suggestions for an alternative to the 812 as a streaming DAC , if the wi-fi dongle solution doesn't work. But with only a wi-fi connection, either to the 812 or to another DAC, you will not get the full benefit from the 812.