From the N-G link you provided: "Some anti-Chief activists deride it as a pseudo-Native American chant." So that will probably be the next target, since it surely offends somebody, somewhere. Fighting Illini can't stay, either. That's just as bad as "Warriors". "Hail to the Orange" is probably safe...no, wait, isn't Orange associated with the Protestant faction in Northern Ireland? The idea that these people are going to be content after getting their latest scalp (pun intended) is bonkers.
You're suggesting that this is a slippery slope and that people on the other side of the debate from you do not care about the facts.
I'm suggesting that most people are willing to listen to the facts and there is in fact a very bright line to separate things to keep from things not to keep: when the tradition was created, was it created in reference to the Chief Illiniwek mascot (or "symbol" for the snowflake PC police we have around here)?
Fighting Illini name -- nope, predates Chief, reference is to students that served in WWI
Hail to the Orange -- no reference to Native Americans
Illinois Loyalty -- official school song from 1906, predates Chief, includes "Oskee Wow Wow" cheer
Oskee Wow Wow -- official fight song from 1911, sounds like it could be problematic, but historical references indicate that it was used as a cheer at Illinois and other schools with no Native American tie
In contrast:
Chief Illiniwek -- originated 1926
War Chant -- not clear when originated but definitely post-Chief and a reference to Chief Illiniwek
March of the Illini -- written specifically for Chief Illiniwek's performances in 1926
Three-In-One routine -- created specifically for Chief Illiniwek's dance
Seems pretty straightforward to me in light of the facts.