>>If you care enough about the quality of the image to calibrate it, why stream in the first place?
That's a good point.
The "streaming input" doesn't change the incoming video (i.e. doesn't make the incoming video too green or reduced gamma, or shifts blue towards green). So, nominally, the "streaming input" should be calibrated to the same standard as all HD inputs... Rec. 709 with white and shades of gray at the xy coordinates of .313/.329 which is the d65 standard coordinates. So Rec 709/d65 is the calibration standard period.
In the days of analog video, every source and every input could be off a little bit from one to another. In digital video, that doesn't happen unless some video processing is applied. So you turn off all the video processing you can and you calibrate to d65 & Rec, 709 and your display is good for any digital video source. If the source is inaccurate (and every YouTube video is different, no way to stop that) your video will be accurate. If the source is inaccurate, you will see the inaccuracies however large or small they are.