That would be the rationalization, although that doesn't really gibe with how it's portrayed on BATWOMAN, with the Bat-Signal being a thing, the mayor declaring that Batman is gone for good, old newspaper headlines about Batman foiling the Joker, etc. Indeed, when Kate first dons the suit, the city's immediate reaction is "He's back!"
As opposed to "He was real?"
Well, Oliver did talk about Batman as an urban myth in
Elseworlds, so that inconsistency was already there. (I assume he was considered a local legend that folks in Gotham believed in but people elsewhere were skeptical of.) But we know that Earth-38 Superman made his debut very publicly with a plane rescue, much as in several other continuities. Then again, maybe his Earth-Prime history is different.
So, let's see... Overall, how many other pre-existing, non-Arrowverse DC film and TV universes had characters appear in
Crisis?
Tim Burton Batman (Earth-89)
Titans (Earth-9)
Batman '66 (Earth-66)
Smallville (Earth-167)
Superman Returns (and Donnerverse indirectly) (Earth-96)
Birds of Prey (Earth-203)
Lucifer (Earth-666)
The Flash '90 (Earth-90)
DCEU (unknown)
Stargirl (Earth-2 -- so maybe it is Arrowverse after all?)
Green Lantern (Earth-12)
Swamp Thing (Earth-19)
Doom Patrol (Earth-21)
By my count, that's 13 in all, though only 5 (two Clark Kents, two Barry Allens, and Lucifer) had more than brief cameos.
So what was left out?
TV:
Adventures of Superman
Shazam/Isis
Wonder Woman '75
Superboy
Swamp Thing '90
Human Target (both '92 and '10)
Lois & Clark
Gotham
Powerless
Krypton
Pennyworth
Watchmen (aside from a billboard)
Film:
Batman, Superman, and other
serials
Swamp Thing
Steel
Constantine (Keanu Reeves version)
Nolanverse
Jonah Hex
(Note: I don't list
Catwoman separately because it's an indirect sequel to
Batman Returns, and thus would have taken place on Earth-89.)
The most surprising omissions are
Lois & Clark and
Gotham. It's also kind of surprising that three of the four ongoing feature continuities (Donner, the two Batman series, and DCEU) got referenced.