I wouldn't consider it a 'downgrade' though. Making him a Thanagarian cop, and introducing the whole mythos behind Thanagar, was a pretty substantial change. In terms of origin story, its a bigger revamp than the Barry Allen Flash was from Jay Garrick's Flash.
Those aren't comparable situations at all. Superman, Batman and Wonder Woman were, for all intents and purposes, completely unchanged! Well, Superman had gradually evolved over the course of the Golden Age into his more 'classic' version, Batman got there within a year of publication, and Wonder Woman had also gradually shifted direction during the transition from Marsdon to Kanigher. But that apart, they were fundamentally the same characters with the same backstories.
Maybe Green Arrow and Speedy are a better point of comparison to the Hawks? Same names, same look, but totally different origin stories in the Silver Age? Or Aquaman - well Aquaman is now going through the opposite of what Hawkman went through in that the Golden Age and Silver Age versions are being split into separate characters, since the only thing they have in common are their powers (mostly) and their looks.
I'll admit I haven't seen the episode in question. If so, I guess this is the first instance of Katar and Carter being merged through reincarnation, like what Venditti did?
Even so, I'd say DCAU Hawkman is baseline the Golden Age/Carter Hall version, with the Thanagarian element thrown in to connect him to Shayera Hol.
Dunno about 'The Batman' either...did they delve into the Thanagarian mythos much there?
There are a lot of times they show Hawkman (or Hawkgirl) as a background character without specifying which version it is on-screen. The DCAMU did something similar.