The topic came up in some other postings in this thread, but rather than keep digging down, I started a new top-level response.
"Calibrated screen" isn't really a thing. A calibrator is used to make a computer/screen
system calibrated. Most, often all, of the actual color correction is done in the computer... mainly in LUTs (Look Up Tables) that are loaded into the computer's video processor. Some more can be done by color-space-aware software running on that computer, which then handles the color-space rendering from the (usually wide-gamut) working space into the LUT-corrected monitor color space.
Sometimes, some monitor adjustments
are made at the start of the process, to bring the monitor closer to the desired response so that the LUTs don't have to make such large changes. With LCD screens, this is less common because LCDs don't always respond well to the changes -- they sometimes do some funny things with color casts.
To the extent that monitor adjustments are made, yes, this can improve the color response from your HDMI input. But it doesn't make the monitor "calibrated."
--
Co-founder of the open-source LightZone Project:
http://lightzoneproject.org/