Yes, Disney did survive 9/11, but not without casualty to stock and personnel.
According to David Koenig in his book "Realityland", after 9/11, they not only cut back on staff and projects, but also on park hours, as well as temporarily shuttered restaurants and had rides operating at the barest possible minimum. Even worse, 9/11 had an impact on Disney stock.
However, the fallout from 9/11 eventually forced the company to (reluctantly, in Koenig's words) to bring out several high-profile attractions (at least at WDW, anyway) to get people to come back. While the guest situation did eventually return to normal, the stock did not. That was one reason why in 2004, Comcast tried to instigate a hostile takeover of Disney and why Roy E. Disney, who was instrumental in bringing in Eisner in 1984, left the company a second time to stage a second coup, this time to get rid of Eisner, who was ultimately replaced by Bob Iger.
So, indirectly at least, 9/11 actually did threaten Disney's existence.
If Disney cuts back, not only will the guests be impacted, but more importantly, so will the stock. Already I've heard about a $200+ million loss caused by the closure of the Chinese parks because of the coronavirus. And while the guest situation might return to normal eventually, the Disney stock may be a different story, just as it was in the period between 9/11 and Eisner's ousting. And now, Roy E. Disney is no longer around to save the company.