Congratulations Jichao, you have prompted the creation of a new word! I think the word that would be most equivalent to "visibility" in this context might be "clusivity" which does not actually exist at this time.
Seeing that invisibility and visibility are antonyms, logically the word inclusivity should have an antonym as well and thus clusivity is born.
Consider if you will that inclusivity is a relatively new word (significant usage begins circa 1930) representing the label for a policy created in response to the elitist concept of exclusivity. My suggestion of course would lead to two different antyonyms for one word but that is not unique in English.
Of course since this is computer related you could also just use boolean logic and choose to use inclusivity for your hypernym, as in inclusivity (state) = true / false. This would not be a new word but could possibly be new usage of the word.
A very interesting question. I approached this from an etymological perspective and found the following (PIE = Proto-Indo-European):
include: from the Latin includere: "to shut in"
from in-: "in" (from PIE root en: "in") + claudere: "to shut / close"
related to enclose (from Old French enclore: "surround / contain / confine")
exclude: from the Latin excludere: "to shut out"
from ex-: "out" (from PIE root eghs: "out") + claudere: "to shut / close"
These in turn led me to explore similar words:
seclude: from the Latin secludere: "to shut off"
from se-: "off / apart" + claudere: "to shut / close"
occlude: from the Latin occludere: "to shut against"
from ob-: "against / in the way of" + claudere: "to shut / close"
The Latin "ob-" is from PIE root epi, opi: "near / at / against"
preclude: from the Latin praecludere: "to shut before"
from pre-, prae-: "before / ahead" + claudere: "to shut / close"
- recluse: from the Latin recludere: "to shut out"
from Latin re-, intensive prefix + claudere: "to shut / close"
After examining these the logical hypernyms for the inclusion/exclusion state could be "shutting", "closing", or (my preference) simply "closure". Unfortunately these terms are not intuitive and would require far too extensive an explanation for most people to grasp.